Download The line y = b is a horizontal asymptote of the graph of a function if

February 10, 2016
Section 1.2d
Functions: Asymptotes
The line y = b is a horizontal asymptote of
the graph of a function if the value of the
function approaches b as a limit as x
approaches ±∞.
Window: [-10, 20] [-10, 12]
Horizontal asymptotes of Rational Functions
1) If the degree of numerator is less than the degree of the denominator
the horizontal asymptote is y = 0.
2) If the degree of numerator is equal to the degree of the denominator
the horizontal asymptote is y = a/b, where a is the leading coefficient of
the numerator and b is the leading coefficient of the denominator.
3) If the degree of the numerator is greater than the degree of the
denominator there is no horizontal asymptote.
What is the maximum number of horizontal
asymptotes a graph may have? Two
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February 10, 2016
The line x = a is a vertical asymptote of the
graph of a function if the value of the function
approaches ±∞ as a limit as x approaches a
from either side.
Window: [-10, 20] [-10, 12]
Vertical asymptotes of Rational Functions
Determine the domain of the function and then evaluate each one-sided
limit for numbers which are not in the domain. A vertical asymptote
exists if the value of any of these limits is ±∞.
What is the maximum number of vertical
asymptotes a graph may have? A function may
have infinitely many vertical asymptotes.
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What is the domain of f(x)?
What is the domain of g(x)?
°
0
0
0
0
-∞
-∞
∞
∞
-0.5
-0.5
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February 10, 2016
What is the domain of g?
What limits must be evaluated to find the
vertical asymptotes.
∞
-∞
-∞
∞
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Use the given limits to help determine the
asymptotes of the function. Then sketch
a possible graph of the function.
Horizontal asymptotes: y = 2 and y = -2
Vertical asymptotes: x = -1 and x = 3
6
4
2
-6
-4
-2
2
4
6
-2
-4
-6
5