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.
5. 2
1
2
x
y
x x
Graph the equation.
1
1 2
x
y
x x
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