Let f be a function defined on some interval (a, ∞). Then
lim f (x) = L
x→∞
means that the values of f (x) can be made arbitrarily close to L by taking x sufficiently large.
Geometric Illustrations:
Let f be a function defined on some interval (−∞, a). Then
lim f (x) = L
x→−∞
means that the values of f (x) can be made arbitrarily close to L by taking x sufficiently large negative.
of the curve y = f (x) if
The line y = L is called a
NOTE: It is possible to have a horizontal asymptote in only one direction or even to have two different
horizontal asymptotes. Thus, it is important when finding horizontal asymptotes that you find the
limit as x approaches ∞ and as x approaches −∞.
1
1
and lim .
x→−∞ x
x→∞ x
Example 16. Find lim
x
1
10
100
10, 000
1, 000, 000
−1
−10
−100
−10, 000
−1, 000, 000
f (x)
x
f (x)
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Most of the Limit Laws that were previously given also hold for limits at infinity.
Important Rules for Calculating Limits:
• If r > 0 is a rational number, then
– lim xr =
x→∞
1
=
x→∞ xr
– lim
• If r > 0 is a rational number such that xr is defined for all x, then
– lim xr =
x→−∞
1
=
x→−∞ xr
– lim
Example 17. Find the limit.
r4 − r2 + 1
a) lim 5
r→∞ r + r 3 − r
√
1 + 4x2
b) lim
x→∞
4+x
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√
1− x
√
c) lim
x→∞ 1 +
x
d) lim (x3 − 5x2 )
x→−∞
x7 − 1
x→∞ x6 + 1
e) lim
Example 18. Find
lim ex
x→∞
lim ex
x→−∞
Example 19. Find the horizontal and vertical asymptotes of each curve.
x
a) y =
x+4
b) y =
x3
x2 + 3x − 10
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