1) Points of discontinuity occur when the denominator of a rational function equals zero.
2) A rational function can have at most one horizontal asymptote. If the degree of the numerator is less than the denominator, the horizontal asymptote is y=0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
3) To find the equation of a slant asymptote, divide the numerator by the denominator and keep only the terms with the highest powers of x.
1. POINTS OF DISCONTINUITY
(Holes and Asymptotes)
Points of discontinuity occur when the denominator
of a function equals zero.
2
2
2
2
1.
2
3
2.
4 3
1
3.
2
x x
y
x
x
y
x x
x
y
x x
1
1 2
x
y
x x
3
3 1
x
y
x x
2 1
2
x x
y
x
2. Horizontal Asymptotes
Note: The graph of a rational function has at most one
horizontal asymptote.
If the degree of the numerator is less than the denominator, the
horizontal asymptote is y = 0.
If the degree of the numerator is more than the denominator, there
is NO horizontal asymptote .
If the degree of the numerator is equal to the degree of the
denominator, the horizontal asymptote is y = a / b , where a and b
are the leading coefficients of the numerator and the denominator.
1
y
x
Example:
2
2
2
2 1
x x
y
x x
Example:
2 1y xExample:
3. EXAMPLE: Finding the Slant/Oblique
Asymptote of a Rational Function
Find the slant asymptotes of f(x)
2
4 5
.
3
x x
x
Solution Because the degree of the numerator, 2, is exactly one
more than the degree of the denominator, 1, the graph of f has a
slant asymptote. To find the equation of the slant asymptote,
divide x 3 into x2 4x 5:
2 1 4 5
1 3 3
1 1 8
3
2
8
1 1
3
3 4 5
x
x
x x x
Remainder
moremore
4. 1 2
1 1
x x
y
x x
2
2
2
1
x x
y
x
Graph the equation.