This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.
This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.
Lesson 27: Integration by Substitution (Section 041 slides)
The document provides notes from a Calculus I class at New York University on December 13, 2010. It includes announcements about upcoming review sessions and the final exam. It then outlines the objectives and topics to be covered, which is integration by substitution. Examples are provided to demonstrate how to use u-substitution to evaluate indefinite integrals. The key steps are to let u equal an expression involving x, find du in terms of dx, and then substitute into the integral.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
The document outlines topics to be covered in Calculus I class sessions on the derivative and rates of change, including: defining the derivative at a point and using it to find the slope of the tangent line to a curve at that point; examples of derivatives modeling rates of change; and how to find the derivative function and second derivative of a given function. It provides learning objectives, an outline of topics, and an example problem worked out graphically and numerically to illustrate finding the slope of the tangent line.
The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
The document is a lecture note on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions like arcsin, arccos, arctan and gives their domains, ranges and other properties. It also provides examples of calculating the values of inverse trig functions like arcsin(1/2) = π/6 and arctan(-1) = -π/4.
This document contains lecture notes on differentiation rules from a Calculus I class at New York University. It begins with objectives to understand basic differentiation rules like the derivative of a constant function, the Constant Multiple Rule, the Sum Rule, and derivatives of sine and cosine. It then provides examples of using the definition of the derivative to find the derivatives of squaring and cubing functions. It illustrates the functions and their derivatives on graphs and discusses properties like a function being increasing when its derivative is positive.
The document discusses optimizing the area of a rectangular field using 320 yards of fencing. It poses the problem of determining the shape of the field that maximizes the enclosed area given the fixed amount of fencing. It also provides a link to an online quiz about derivatives and graphing that contains problems requiring deep thought about concepts learned.
This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It covers objectives like differentiating constant, sum, and difference functions. It also reviews derivatives of sine and cosine. Examples are provided, like finding the derivative of a squaring function x^2 using the definition of a derivative. The document outlines the topics and provides explanations and step-by-step solutions.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
This lecture covers various solution methods for unconstrained optimization problems, including:
1) Line search methods like dichotomous search and the Fibonacci and golden section methods for one-dimensional problems.
2) Newton's method and the false position method for curve fitting to minimize functions in one dimension.
3) Descent methods for multidimensional problems like the gradient method, which follows the negative gradient direction at each step, and how scaling can improve convergence rates.
This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.
This document provides an overview of L'Hopital's rule for evaluating limits of indeterminate forms. It begins by defining different types of indeterminate forms such as 0/0, infinity/infinity, 0*infinity, infinity - infinity, and 00. It then introduces L'Hopital's rule, which allows such limits to be evaluated by taking the derivative of the numerator and denominator. Several examples are worked out to demonstrate how L'Hopital's rule can be applied. The document concludes by discussing various types of relative growth rates between functions as x approaches infinity.
This document provides an overview of integration by substitution. It begins with announcements about an upcoming review session, evaluations, and final exam. It then discusses the objectives and outline of the section, which are to transform integrals using substitutions, evaluate indefinite integrals using substitutions, and evaluate definite integrals using substitutions. An example is provided to illustrate how to use substitution to evaluate the indefinite integral of x/(x^2 + 1) by letting u = x^2 + 1. The solution uses a new notation of letting u = x^2 + 1 and du = 2x dx to rewrite the integral in terms of u.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. Examples are provided of finding the antiderivative of power functions like x^3 through identifying the power rule relationship between a function and its derivative.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.
Lesson 14: Derivatives of Logarithmic and Exponential Functions
The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.
This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of
- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals.
- It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
This document outlines information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences. It provides details on the course staff, contact information for the professor, an overview of assessments including homework, quizzes, a midterm and final exam. Grading breakdown is also included, as well as information on purchasing the required textbook and accessing the course on Blackboard. The document aims to provide students with essential logistical information to succeed in the Calculus I course.
This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It includes announcements about homework and deadlines. It then discusses guidelines for written homework assignments and a rubric for grading. The bulk of the document explains the concept of limit intuitively through an "error-tolerance game" and provides examples to illustrate the idea. It aims to build an informal understanding of limits before providing a precise definition.
This document is a section from a Calculus I course at New York University dated September 20, 2010. It discusses continuity of functions, beginning with definitions and examples of determining continuity. Key points covered include the definition of continuity as a function having a limit equal to its value, and theorems stating polynomials, rational functions, and combinations of continuous functions are also continuous. Trigonometric functions like sin, cos, tan, and cot are shown to be continuous on their domains. The document provides examples, explanations, and questions to illustrate the concept of continuity.
This document provides an overview of calculating limits from a Calculus I course at New York University. It begins with announcements about homework being due and includes sections on recalling the concept of limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The document uses examples, definitions, proofs, and the error-tolerance game to explain how to calculate limits, limit laws, and applying algebra to limits. It provides the essential information and objectives for understanding how to compute elementary limits.
This document is from a Calculus I course at New York University and covers limits involving infinity. It discusses definitions of limits approaching positive or negative infinity. Examples are provided of functions with infinite limits, such as 1/x as x approaches 0. The document outlines techniques for finding limits at points where a function is not continuous, such as using a number line to determine the signs of factors in a rational function's denominator.
This document is from a Calculus I course at New York University and covers limits involving infinity. It defines infinite limits, both limits approaching positive and negative infinity, and limits at vertical asymptotes. Examples are provided of known infinite limits, like limits of 1/x as x approaches 0. The document demonstrates finding one-sided limits at points where a function is not continuous using a number line to determine the sign of factors in the denominator.
Why do You Have to Redesign?_Redesign Challenge Day 1
Are you interested in learning about creating an attractive website? Here it is! Take part in the challenge that will broaden your knowledge about creating cool websites! Don't miss this opportunity, only in "Redesign Challenge"!
Blockchain technology is transforming industries and reshaping the way we conduct business, manage data, and secure transactions. Whether you're new to blockchain or looking to deepen your knowledge, our guidebook, "Blockchain for Dummies", is your ultimate resource.
How Social Media Hackers Help You to See Your Wife's Message.pdf
In the modern digital era, social media platforms have become integral to our daily lives. These platforms, including Facebook, Instagram, WhatsApp, and Snapchat, offer countless ways to connect, share, and communicate.
This document is from a Calculus I course at New York University. It outlines the objectives and content for Sections 2.1-2.2 on the derivative and rates of change. The sections will define the derivative, discuss how it models rates of change, and how to find the derivative of various functions graphically and numerically. It will also cover how to find higher-order derivatives and sketch the derivative graph from a function graph.
This document is from a Calculus I class at New York University and covers continuity. It provides announcements about office hours and homework deadlines. It then discusses the objectives of understanding the definition of continuity and applying it to piecewise functions. Examples are provided to demonstrate how to show a function is continuous at a point by evaluating the limit as x approaches the point and showing it equals the function value. The students are asked to determine at which other points the example function is continuous.
Presentation deck used at the Model Schools Conference in Orlando 2012. Presentation on KP Compass and how we use game theory to increase student engagement in our concept driven mastery system. www.kpcompass.com
This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.
Lesson 27: Integration by Substitution (Section 041 slides)Mel Anthony Pepito
The document provides notes from a Calculus I class at New York University on December 13, 2010. It includes announcements about upcoming review sessions and the final exam. It then outlines the objectives and topics to be covered, which is integration by substitution. Examples are provided to demonstrate how to use u-substitution to evaluate indefinite integrals. The key steps are to let u equal an expression involving x, find du in terms of dx, and then substitute into the integral.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
The document outlines topics to be covered in Calculus I class sessions on the derivative and rates of change, including: defining the derivative at a point and using it to find the slope of the tangent line to a curve at that point; examples of derivatives modeling rates of change; and how to find the derivative function and second derivative of a given function. It provides learning objectives, an outline of topics, and an example problem worked out graphically and numerically to illustrate finding the slope of the tangent line.
The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
Lesson 16: Inverse Trigonometric Functions (Section 041 slides)Mel Anthony Pepito
The document is a lecture note on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions like arcsin, arccos, arctan and gives their domains, ranges and other properties. It also provides examples of calculating the values of inverse trig functions like arcsin(1/2) = π/6 and arctan(-1) = -π/4.
This document contains lecture notes on differentiation rules from a Calculus I class at New York University. It begins with objectives to understand basic differentiation rules like the derivative of a constant function, the Constant Multiple Rule, the Sum Rule, and derivatives of sine and cosine. It then provides examples of using the definition of the derivative to find the derivatives of squaring and cubing functions. It illustrates the functions and their derivatives on graphs and discusses properties like a function being increasing when its derivative is positive.
The document discusses optimizing the area of a rectangular field using 320 yards of fencing. It poses the problem of determining the shape of the field that maximizes the enclosed area given the fixed amount of fencing. It also provides a link to an online quiz about derivatives and graphing that contains problems requiring deep thought about concepts learned.
This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It covers objectives like differentiating constant, sum, and difference functions. It also reviews derivatives of sine and cosine. Examples are provided, like finding the derivative of a squaring function x^2 using the definition of a derivative. The document outlines the topics and provides explanations and step-by-step solutions.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
This lecture covers various solution methods for unconstrained optimization problems, including:
1) Line search methods like dichotomous search and the Fibonacci and golden section methods for one-dimensional problems.
2) Newton's method and the false position method for curve fitting to minimize functions in one dimension.
3) Descent methods for multidimensional problems like the gradient method, which follows the negative gradient direction at each step, and how scaling can improve convergence rates.
This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.
This document provides an overview of L'Hopital's rule for evaluating limits of indeterminate forms. It begins by defining different types of indeterminate forms such as 0/0, infinity/infinity, 0*infinity, infinity - infinity, and 00. It then introduces L'Hopital's rule, which allows such limits to be evaluated by taking the derivative of the numerator and denominator. Several examples are worked out to demonstrate how L'Hopital's rule can be applied. The document concludes by discussing various types of relative growth rates between functions as x approaches infinity.
This document provides an overview of integration by substitution. It begins with announcements about an upcoming review session, evaluations, and final exam. It then discusses the objectives and outline of the section, which are to transform integrals using substitutions, evaluate indefinite integrals using substitutions, and evaluate definite integrals using substitutions. An example is provided to illustrate how to use substitution to evaluate the indefinite integral of x/(x^2 + 1) by letting u = x^2 + 1. The solution uses a new notation of letting u = x^2 + 1 and du = 2x dx to rewrite the integral in terms of u.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. Examples are provided of finding the antiderivative of power functions like x^3 through identifying the power rule relationship between a function and its derivative.
Similar to Lesson 17: Indeterminate Forms and L'Hôpital's Rule (8)
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMel Anthony Pepito
The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.
This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of
- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals.
- It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
This document outlines information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences. It provides details on the course staff, contact information for the professor, an overview of assessments including homework, quizzes, a midterm and final exam. Grading breakdown is also included, as well as information on purchasing the required textbook and accessing the course on Blackboard. The document aims to provide students with essential logistical information to succeed in the Calculus I course.
This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It includes announcements about homework and deadlines. It then discusses guidelines for written homework assignments and a rubric for grading. The bulk of the document explains the concept of limit intuitively through an "error-tolerance game" and provides examples to illustrate the idea. It aims to build an informal understanding of limits before providing a precise definition.
This document is a section from a Calculus I course at New York University dated September 20, 2010. It discusses continuity of functions, beginning with definitions and examples of determining continuity. Key points covered include the definition of continuity as a function having a limit equal to its value, and theorems stating polynomials, rational functions, and combinations of continuous functions are also continuous. Trigonometric functions like sin, cos, tan, and cot are shown to be continuous on their domains. The document provides examples, explanations, and questions to illustrate the concept of continuity.
This document provides an overview of calculating limits from a Calculus I course at New York University. It begins with announcements about homework being due and includes sections on recalling the concept of limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The document uses examples, definitions, proofs, and the error-tolerance game to explain how to calculate limits, limit laws, and applying algebra to limits. It provides the essential information and objectives for understanding how to compute elementary limits.
This document is from a Calculus I course at New York University and covers limits involving infinity. It discusses definitions of limits approaching positive or negative infinity. Examples are provided of functions with infinite limits, such as 1/x as x approaches 0. The document outlines techniques for finding limits at points where a function is not continuous, such as using a number line to determine the signs of factors in a rational function's denominator.
This document is from a Calculus I course at New York University and covers limits involving infinity. It defines infinite limits, both limits approaching positive and negative infinity, and limits at vertical asymptotes. Examples are provided of known infinite limits, like limits of 1/x as x approaches 0. The document demonstrates finding one-sided limits at points where a function is not continuous using a number line to determine the sign of factors in the denominator.
Are you interested in learning about creating an attractive website? Here it is! Take part in the challenge that will broaden your knowledge about creating cool websites! Don't miss this opportunity, only in "Redesign Challenge"!
Blockchain technology is transforming industries and reshaping the way we conduct business, manage data, and secure transactions. Whether you're new to blockchain or looking to deepen your knowledge, our guidebook, "Blockchain for Dummies", is your ultimate resource.
How Social Media Hackers Help You to See Your Wife's Message.pdfHackersList
In the modern digital era, social media platforms have become integral to our daily lives. These platforms, including Facebook, Instagram, WhatsApp, and Snapchat, offer countless ways to connect, share, and communicate.
How Netflix Builds High Performance Applications at Global ScaleScyllaDB
We all want to build applications that are blazingly fast. We also want to scale them to users all over the world. Can the two happen together? Can users in the slowest of environments also get a fast experience? Learn how we do this at Netflix: how we understand every user's needs and preferences and build high performance applications that work for every user, every time.
7 Most Powerful Solar Storms in the History of Earth.pdfEnterprise Wired
Solar Storms (Geo Magnetic Storms) are the motion of accelerated charged particles in the solar environment with high velocities due to the coronal mass ejection (CME).
An invited talk given by Mark Billinghurst on Research Directions for Cross Reality Interfaces. This was given on July 2nd 2024 as part of the 2024 Summer School on Cross Reality in Hagenberg, Austria (July 1st - 7th)
What's Next Web Development Trends to Watch.pdfSeasiaInfotech2
Explore the latest advancements and upcoming innovations in web development with our guide to the trends shaping the future of digital experiences. Read our article today for more information.
Details of description part II: Describing images in practice - Tech Forum 2024BookNet Canada
This presentation explores the practical application of image description techniques. Familiar guidelines will be demonstrated in practice, and descriptions will be developed “live”! If you have learned a lot about the theory of image description techniques but want to feel more confident putting them into practice, this is the presentation for you. There will be useful, actionable information for everyone, whether you are working with authors, colleagues, alone, or leveraging AI as a collaborator.
Link to presentation recording and transcript: https://bnctechforum.ca/sessions/details-of-description-part-ii-describing-images-in-practice/
Presented by BookNet Canada on June 25, 2024, with support from the Department of Canadian Heritage.
Video traffic on the Internet is constantly growing; networked multimedia applications consume a predominant share of the available Internet bandwidth. A major technical breakthrough and enabler in multimedia systems research and of industrial networked multimedia services certainly was the HTTP Adaptive Streaming (HAS) technique. This resulted in the standardization of MPEG Dynamic Adaptive Streaming over HTTP (MPEG-DASH) which, together with HTTP Live Streaming (HLS), is widely used for multimedia delivery in today’s networks. Existing challenges in multimedia systems research deal with the trade-off between (i) the ever-increasing content complexity, (ii) various requirements with respect to time (most importantly, latency), and (iii) quality of experience (QoE). Optimizing towards one aspect usually negatively impacts at least one of the other two aspects if not both. This situation sets the stage for our research work in the ATHENA Christian Doppler (CD) Laboratory (Adaptive Streaming over HTTP and Emerging Networked Multimedia Services; https://athena.itec.aau.at/), jointly funded by public sources and industry. In this talk, we will present selected novel approaches and research results of the first year of the ATHENA CD Lab’s operation. We will highlight HAS-related research on (i) multimedia content provisioning (machine learning for video encoding); (ii) multimedia content delivery (support of edge processing and virtualized network functions for video networking); (iii) multimedia content consumption and end-to-end aspects (player-triggered segment retransmissions to improve video playout quality); and (iv) novel QoE investigations (adaptive point cloud streaming). We will also put the work into the context of international multimedia systems research.
UiPath Community Day Kraków: Devs4Devs ConferenceUiPathCommunity
We are honored to launch and host this event for our UiPath Polish Community, with the help of our partners - Proservartner!
We certainly hope we have managed to spike your interest in the subjects to be presented and the incredible networking opportunities at hand, too!
Check out our proposed agenda below 👇👇
08:30 ☕ Welcome coffee (30')
09:00 Opening note/ Intro to UiPath Community (10')
Cristina Vidu, Global Manager, Marketing Community @UiPath
Dawid Kot, Digital Transformation Lead @Proservartner
09:10 Cloud migration - Proservartner & DOVISTA case study (30')
Marcin Drozdowski, Automation CoE Manager @DOVISTA
Pawel Kamiński, RPA developer @DOVISTA
Mikolaj Zielinski, UiPath MVP, Senior Solutions Engineer @Proservartner
09:40 From bottlenecks to breakthroughs: Citizen Development in action (25')
Pawel Poplawski, Director, Improvement and Automation @McCormick & Company
Michał Cieślak, Senior Manager, Automation Programs @McCormick & Company
10:05 Next-level bots: API integration in UiPath Studio (30')
Mikolaj Zielinski, UiPath MVP, Senior Solutions Engineer @Proservartner
10:35 ☕ Coffee Break (15')
10:50 Document Understanding with my RPA Companion (45')
Ewa Gruszka, Enterprise Sales Specialist, AI & ML @UiPath
11:35 Power up your Robots: GenAI and GPT in REFramework (45')
Krzysztof Karaszewski, Global RPA Product Manager
12:20 🍕 Lunch Break (1hr)
13:20 From Concept to Quality: UiPath Test Suite for AI-powered Knowledge Bots (30')
Kamil Miśko, UiPath MVP, Senior RPA Developer @Zurich Insurance
13:50 Communications Mining - focus on AI capabilities (30')
Thomasz Wierzbicki, Business Analyst @Office Samurai
14:20 Polish MVP panel: Insights on MVP award achievements and career profiling
Kief Morris rethinks the infrastructure code delivery lifecycle, advocating for a shift towards composable infrastructure systems. We should shift to designing around deployable components rather than code modules, use more useful levels of abstraction, and drive design and deployment from applications rather than bottom-up, monolithic architecture and delivery.
3. Objectives
Know when a limit is of
indeterminate form:
indeterminate quotients:
0/0, ∞/∞
indeterminate products:
0×∞
indeterminate differences:
∞−∞
indeterminate powers: 00 ,
∞0 , and 1∞
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 3 / 26
4. Experiments with funny limits
sin2 x
lim
x→0 x
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 4 / 26
5. Experiments with funny limits
sin2 x
lim =0
x→0 x
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 4 / 26
6. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim
x→0 sin2 x
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 4 / 26
7. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 4 / 26
8. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
sin2 x
lim
x→0 sin(x 2 )
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 4 / 26
9. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
sin2 x
lim =1
x→0 sin(x 2 )
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 4 / 26
10. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
sin2 x
lim =1
x→0 sin(x 2 )
sin 3x
lim
x→0 sin x
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 4 / 26
11. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
sin2 x
lim =1
x→0 sin(x 2 )
sin 3x
lim =3
x→0 sin x
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 4 / 26
12. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
sin2 x
lim =1
x→0 sin(x 2 )
sin 3x
lim =3
x→0 sin x
0
All of these are of the form , and since we can get different answers in
0
different cases, we say this form is indeterminate.
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 4 / 26
13. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 5 / 26
14. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
Limit of a difference is the difference of the limits
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 5 / 26
15. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
Limit of a difference is the difference of the limits
Limit of a product is the product of the limits
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 5 / 26
16. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
Limit of a difference is the difference of the limits
Limit of a product is the product of the limits
Limit of a quotient is the quotient of the limits ... whoops! This is
true as long as you don’t try to divide by zero.
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 5 / 26
17. More about dividing limits
We know dividing by zero is bad.
Most of the time, if an expression’s numerator approaches a finite
number and denominator approaches zero, the quotient approaches
some kind of infinity. For example:
1 cos x
lim+ = +∞ lim = −∞
x→0 x x→0− x3
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 6 / 26
18. More about dividing limits
We know dividing by zero is bad.
Most of the time, if an expression’s numerator approaches a finite
number and denominator approaches zero, the quotient approaches
some kind of infinity. For example:
1 cos x
lim+ = +∞ lim = −∞
x→0 x x→0− x3
An exception would be something like
1
lim = lim x csc x.
x→∞ 1 sin x x→∞
x
which doesn’t exist.
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 6 / 26
19. More about dividing limits
We know dividing by zero is bad.
Most of the time, if an expression’s numerator approaches a finite
number and denominator approaches zero, the quotient approaches
some kind of infinity. For example:
1 cos x
lim+ = +∞ lim = −∞
x→0 x x→0− x3
An exception would be something like
1
lim = lim x csc x.
x→∞ 1 sin x x→∞
x
which doesn’t exist.
Even less predictable: numerator and denominator both go to zero.
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 6 / 26
20. Language Note
It depends on what the meaning of the word “is” is
Be careful with the language
here. We are not saying that
the limit in each case “is”
0
, and therefore nonexistent
0
because this expression is
undefined.
0
The limit is of the form ,
0
which means we cannot
evaluate it with our limit
laws.
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 7 / 26
21. Indeterminate forms are like Tug Of War
Which side wins depends on which side is stronger.
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 8 / 26
22. Outline
L’Hˆpital’s Rule
o
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
Summary
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 9 / 26
23. The Linear Case
Question
If f and g are lines and f (a) = g (a) = 0, what is
f (x)
lim ?
x→a g (x)
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 10 / 26
24. The Linear Case
Question
If f and g are lines and f (a) = g (a) = 0, what is
f (x)
lim ?
x→a g (x)
Solution
The functions f and g can be written in the form
f (x) = m1 (x − a)
g (x) = m2 (x − a)
So
f (x) m1
=
g (x) m2
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 10 / 26
25. The Linear Case, Illustrated
y y = g (x)
y = f (x)
g (x)
a f (x)
x
x
f (x) f (x) − f (a) (f (x) − f (a))/(x − a) m1
= = =
g (x) g (x) − g (a) (g (x) − g (a))/(x − a) m2
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 11 / 26
26. What then?
But what if the functions aren’t linear?
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 12 / 26
27. What then?
But what if the functions aren’t linear?
Can we approximate a function near a point with a linear function?
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 12 / 26
28. What then?
But what if the functions aren’t linear?
Can we approximate a function near a point with a linear function?
What would be the slope of that linear function?
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 12 / 26
29. What then?
But what if the functions aren’t linear?
Can we approximate a function near a point with a linear function?
What would be the slope of that linear function? The derivative!
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 12 / 26
30. Theorem of the Day
Theorem (L’Hopital’s Rule)
Suppose f and g are differentiable functions and g (x) = 0 near a (except
possibly at a). Suppose that
lim f (x) = 0 and lim g (x) = 0
x→a x→a
or
lim f (x) = ±∞ and lim g (x) = ±∞
x→a x→a
Then
f (x) f (x)
lim = lim ,
x→a g (x) x→a g (x)
if the limit on the right-hand side is finite, ∞, or −∞.
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 13 / 26
31. Meet the Mathematician: L’Hˆpital
o
wanted to be a military
man, but poor eyesight
forced him into math
did some math on his own
(solved the “brachistocrone
problem”)
paid a stipend to Johann
Bernoulli, who proved this
theorem and named it after
him! Guillaume Fran¸ois Antoine,
c
Marquis de L’Hˆpital
o
(French, 1661–1704)
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 14 / 26
32. Revisiting the previous examples
Example
sin2 x
lim
x→0 x
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
33. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim
x→0 x x→0 1
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
34. Revisiting the previous examples
Example sin x → 0
sin2 x H 2 sin x cos x
lim = lim
x→0 x x→0 1
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
35. Revisiting the previous examples
Example sin x → 0
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
36. Revisiting the previous examples
Example sin x → 0
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
sin2 x
lim
x→0 sin x 2
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
37. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example numerator → 0
sin2 x
lim
x→0 sin x 2
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
38. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example numerator → 0
sin2 x
lim
x→0 sin x 2
denominator → 0
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
39. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example numerator → 0
sin2 x H 2 sin x cos x
¡
lim 2
= lim
x→0 sin x x→0 (cos x 2 ) (2x )
¡
denominator → 0
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
40. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example numerator → 0
sin2 x H 2 sin x cos x
¡
lim 2
= lim
x→0 sin x x→0 (cos x 2 ) (2x )
¡
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
41. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example numerator → 0
sin2 x H 2 sin x cos x
¡
lim 2
= lim
x→0 sin x x→0 (cos x 2 ) (2x )
¡
denominator → 0
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
42. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example numerator → 0
sin2 x H 2 sin x cos x H
¡ cos2 x − sin2 x
lim = lim = lim
x→0 sin x 2 x→0 (cos x 2 ) (2x )
¡ x→0 cos x 2 − 2x 2 sin(x 2 )
denominator → 0
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
43. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example numerator → 1
sin2 x H 2 sin x cos x H
¡ cos2 x − sin2 x
lim = lim = lim
x→0 sin x 2 x→0 (cos x 2 ) (2x )
¡ x→0 cos x 2 − 2x 2 sin(x 2 )
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
44. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example numerator → 1
sin2 x H 2 sin x cos x H
¡ cos2 x − sin2 x
lim = lim = lim
x→0 sin x 2 x→0 (cos x 2 ) (2x )
¡ x→0 cos x 2 − 2x 2 sin(x 2 )
denominator → 1
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
45. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
sin2 x H 2 sin x cos x H
¡ cos2 x − sin2 x
lim = lim = lim =1
x→0 sin x 2 x→0 (cos x 2 ) (2x )
¡ x→0 cos x 2 − 2x 2 sin(x 2 )
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
46. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
sin2 x H 2 sin x cos x H
¡ cos2 x − sin2 x
lim = lim = lim =1
x→0 sin x 2 x→0 (cos x 2 ) (2x )
¡ x→0 cos x 2 − 2x 2 sin(x 2 )
Example
sin 3x H 3 cos 3x
lim = lim = 3.
x→0 sin x x→0 cos x
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 15 / 26
47. Another Example
Example
Find
x
lim
x→0 cos x
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 16 / 26
48. Beware of Red Herrings
Example
Find
x
lim
x→0 cos x
Solution
The limit of the denominator is 1, not 0, so L’Hˆpital’s rule does not
o
apply. The limit is 0.
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 16 / 26
49. Outline
L’Hˆpital’s Rule
o
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
Summary
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 17 / 26
50. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 18 / 26
51. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then apply
L’Hˆpital’s Rule:
o
√
lim x ln x
x→0+
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 18 / 26
52. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then apply
L’Hˆpital’s Rule:
o
√ ln x
lim x ln x = lim+ 1/√x
x→0+ x→0
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 18 / 26
53. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then apply
L’Hˆpital’s Rule:
o
√ ln x H x −1
lim x ln x = lim+ √ = lim+
x→0+ x→0 1/ x x→0 − 1 x −3/2
2
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 18 / 26
54. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then apply
L’Hˆpital’s Rule:
o
√ ln x H x −1
lim x ln x = lim+ √ = lim+
x→0+ x→0 1/ x x→0 − 1 x −3/2
2
√
= lim+ −2 x
x→0
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 18 / 26
55. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then apply
L’Hˆpital’s Rule:
o
√ ln x H x −1
lim x ln x = lim+ √ = lim+
x→0+ x→0 1/ x x→0 − 1 x −3/2
2
√
= lim+ −2 x = 0
x→0
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 18 / 26
56. Indeterminate differences
Example
1
lim+ − cot 2x
x→0 x
This limit is of the form ∞ − ∞.
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 19 / 26
57. Indeterminate differences
Example
1
lim+ − cot 2x
x→0 x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x)
lim
x→0+ x sin(2x)
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 19 / 26
58. Indeterminate differences
Example
1
lim+ − cot 2x
x→0 x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x)
lim = lim+
x→0+ x sin(2x) x→0 2x cos(2x) + sin(2x)
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 19 / 26
59. Indeterminate differences
Example
1
lim+ − cot 2x
x→0 x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x)
lim = lim+
x→0+ x sin(2x) x→0 2x cos(2x) + sin(2x)
=∞
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 19 / 26
60. Indeterminate differences
Example
1
lim+ − cot 2x
x→0 x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x)
lim = lim+
x→0+ x sin(2x) x→0 2x cos(2x) + sin(2x)
=∞
The limit is +∞ becuase the numerator tends to 1 while the denominator
tends to zero but remains positive.
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 19 / 26
61. Checking your work
tan 2x
lim = 1, so for small x,
x→0 2x
1
tan 2x ≈ 2x. So cot 2x ≈ and
2x
1 1 1 1
− cot 2x ≈ − = →∞
x x 2x 2x
as x → 0+ .
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 20 / 26
62. Indeterminate powers
Example
Find lim+ (1 − 2x)1/x
x→0
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 21 / 26
63. Indeterminate powers
Example
Find lim+ (1 − 2x)1/x
x→0
Take the logarithm:
ln(1 − 2x)
ln lim+ (1 − 2x)1/x = lim+ ln (1 − 2x)1/x = lim+
x→0 x→0 x→0 x
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 21 / 26
64. Indeterminate powers
Example
Find lim+ (1 − 2x)1/x
x→0
Take the logarithm:
ln(1 − 2x)
ln lim+ (1 − 2x)1/x = lim+ ln (1 − 2x)1/x = lim+
x→0 x→0 x→0 x
0
This limit is of the form , so we can use L’Hˆpital:
o
0
−2
ln(1 − 2x) H 1−2x
lim+ = lim+ = −2
x→0 x x→0 1
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 21 / 26
65. Indeterminate powers
Example
Find lim+ (1 − 2x)1/x
x→0
Take the logarithm:
ln(1 − 2x)
ln lim+ (1 − 2x)1/x = lim+ ln (1 − 2x)1/x = lim+
x→0 x→0 x→0 x
0
This limit is of the form , so we can use L’Hˆpital:
o
0
−2
ln(1 − 2x) H 1−2x
lim+ = lim+ = −2
x→0 x x→0 1
This is not the answer, it’s the log of the answer! So the answer we want
is e −2 .
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 21 / 26
66. Another indeterminate power limit
Example
lim (3x)4x
x→0
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 22 / 26
67. Another indeterminate power limit
Example
lim (3x)4x
x→0
Solution
ln lim+ (3x)4x = lim+ ln(3x)4x = lim+ 4x ln(3x)
x→0 x→0 x→0
ln(3x) H 3/3x
= lim+ 1/4x
= lim+ −1/4x 2
x→0 x→0
= lim+ (−4x) = 0
x→0
So the answer is e 0 = 1.
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 22 / 26
68. Summary
Form Method
0
0 L’Hˆpital’s rule directly
o
∞
∞ L’Hˆpital’s rule directly
o
0 ∞
0·∞ jiggle to make 0 or ∞.
∞−∞ factor to make an indeterminate product
00 take ln to make an indeterminate product
∞0 ditto
1∞ ditto
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 23 / 26
69. Final Thoughts
L’Hˆpital’s Rule only works on indeterminate quotients
o
Luckily, most indeterminate limits can be transformed into
indeterminate quotients
L’Hˆpital’s Rule gives wrong answers for non-indeterminate limits!
o
V63.0121.002.2010Su, Calculus I (NYU) L’Hˆpital’s Rule
o June 7, 2010 24 / 26