Asymptotes
Definition of a Limit at Infinity
Definition
of a Limit at Infinity
Let
L
be a real number and
f(x)
be a function. Then
if
for every e >
0, there is an M > 0 such that
f(x)  L < e
whenever x > M. 
In other words as x gets
very large f(x) gets very close to
L.
If
then
we say that y = L is a horizontal asymptote of
f(x).
Example
Find
the horizontal asymptote of
x^{2}  1
f(x) =
2x^{2} + x  3
Solution:
Divide
by x^{2} on the numerator and the denominator to get
Horizontal Asymptotes of NonRational Functions
Example
Find the horizontal asymptotes of
Solution
We must consider the negative infinity case separately from the positive
infinity case. First note that for negative x,
hence
Next for positive,
hence
We
see that there is a left horizontal asymptote at y = 1/2 and a right
horizontal asymptote at y = 1/2.
Example
Find the horizontal asymptotes of
sin x
f (x) =
x
Solution
We see that
1 sin
x 1

<
<
x
x
x
for all x. Both of the outer limits approach
0 as x approaches either
infinity or negative infinity. By the squeeze theorem, the middle
limit must approach zero. We can conclude that f has a horizontal
asymptote at y = 0.
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