Calculus Help
Name: Limits and Asymptotes Section:
4.6 Limits and Asymptotes Vocabulary Examples
Limit at Infinity If lim
x→∞ approaches L��, then L can be referred to as a
at
If lim x→−∞
approaches L��, then L can be referred to as a
at Horizontal Asymptote For a function f , the line y = where lim
x→∞ f (x) = or
lim x→−∞
f (x) = Infinite Limit At Infinity
For a function f , if lim x→∞
= , then f has an infinite limit at infinity.
1. Graph the following rational functions. Label all critical points and asymptotes.
(a) f (x) = 2x 2
x2+x−6 (b) f (x) = x
3
2x2−10
2. Evaluate the following limits.
(a) lim x→∞
3x2−6x5 2x5+3x2−1
(b) lim x→∞
3x √
x2+1 (c) lim
x→−∞ 3x √
x2+1
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Name: Limits and Asymptotes Section:
3. Evaluate the following limits.
(a) lim x→∞
sin x+cos x sin x−cos x (b) limx→∞
10 cos x x (c) limx→∞
1−5ex ex
4. Evaluate the following limits.
(a) lim x→−∞
3x5+x4 100x4−1
(b) lim x→−∞
x2 √
1−x (c) lim
x→−∞ 101x ln(−x)+20x2
4x2+1
5. If f ′(x) has asymptotes at y = 3 and x = 1, then f (x) has what asymptotes? Sketch a possible graph for f (x) and f ′(x) on the same set of axes.
6. Determine the oblique asymptotes of f (x) = √
x2 − 5x + 4
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 55
Name: L’Hopital’s Rule Section:
4.8 L’Hopital’s Rule Vocabulary Examples
L’Hopital’s Rule f and g are differentiable functions over an open interval
containing a. If lim x→a
f (x) = and lim x→a
g(x) = , then
lim x→a
f (x) g(x) =
In addition, if lim x→a
f (x) = and lim x→a
g(x) = , then
lim x→a
f (x) g(x) =
1. Determine whether L’Hopital’s Rule can be applied to each of the following limits.
(a) lim x→∞
ex x
(b) lim x→∞
x ln x (c) lim x→1
x2 x−2
(d) lim x→0
1−cos x x
Determine the following limits.
2. lim x→0
sin x x 3. limx→1
ln x x−1 4. limx→∞
ex x2
5. lim x→π
sin x x 6. limx→0
1−cos x x 7. limx→−∞
cos(ex) ex
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 60
Name: L’Hopital’s Rule Section:
Determine the following limits.
8. lim x→π
π−x sin x 9. limx→π2
sec x 1+tan x 10. lim
x→3 xex x−2
11. lim x→0
(1+x)n−1 x
12. lim x→0
sin x−tan x x3
13. lim x→∞
ln x x3
14. lim x→0
√ 1+x−
√ 1−x
x 15. lim
x→0 ex−x−1
x2 16. lim
x→0 tan x √
x
17. Evaluate the following limits for a , 0
(a) lim x→a
x−a x2−a2
(b) lim x→a
x−a x3−a3
(c) lim x→a
x−a xn−an
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