L’Hôpital’s rule practice problems
21-121: Integration and Differential Equations
Find the following limits. You may use L’Hôpital’s rule where appropriate. Be aware that L’Hôpital’s
rule may not apply to every limit, and it may not be helpful even when it does apply. Some
limits may be found by other methods. These problems are given in no particular order. (Where
appropriate, sources for the problems are given in square brackets under the answer. See the end
for an explanation of these references.)
1
ex − x − 1
19. lim x sin
Ans. 0.
1. lim
Ans. −1.
x→0
x→0 cos x − 1
x
[ R 4.7 Ex. 4 ]
5
tan 3x
Ans. .
20. lim
3
x3 − x2 − 10x − 8
tan
5x
3
x→π/2
Ans. .
2. lim
x→−2 5x3 + 12x2 − 2x − 12
5
[ R 4.7.18 ]
x−a
sec
x
Ans. a.
3. lim
21. lim
Ans. −∞.
x→a ln x − ln a
x→π/2+ ln sec x
[ SSJ 124.29 ]
[ M 62.26 ]
1 − cos θ
1
1
4. lim
Ans. .
Ans. 0.
22. lim csc x −
θ→0
θ2
2
x→0
x
[ M 62.12 ]
5. lim (sinh x − cosh x)
x→∞
6.
tan x
ln(2x − π)
lim +
x→π/2
tanh x
x→∞ tan−1 x
sin x
8. lim
x→0 sinh x
x
e
1
9. lim
−
x→0 ex − 1
x
7. lim
Ans. 0.
23.
Ans. ∞.
24.
lim (tan x)sin 2x
lim x−3 ex
2
.
π
25. lim
Ans. 1.
26. lim
Ans.
1
.
2
[ S 21.3(b) ]
Ans. 0.
x→−∞
[ W VIII.2.1 ]
Ans.
Ans. 1.
x→π/2
ln(t + 2)
log2 t
x→∞
Ans. ln 2.
[ R Ch. 7 Rev. 115 ]
1 − ln x
x/e − 1
√
√
1 − tan x − 1 + tan x
27. lim
x→π
sin 2x
x→e
[ R Ch. 4 rev. 116 ]
10.
lim
x→π/2
ln(x − π/2)
sec x
14. lim
x→∞
15. lim
x→∞
e−x
sin x
e−x
sin x + 2
tan−1 x − π4
16. lim
x→1 tan π x − 1
4
17.
lim
x→−∞
tan−1 x
cot−1 x
Ans. 0.
[ SSJ 122 Ex. 4 ]
Ans. 4e4 .
[ R 4.7.22 ]
Ans. ∞.
Ans.
√
x→0
[ Sh 9.1.14 ]
(π/2) − tan x
x−1
√
x + 10 + 3x1/3
29. lim
x→−1
4x2 + 3x − 1
sin x − x
30. lim
x→0
x3
28. lim
x→∞
Ans. 1.
[ W VIII.1.3 Ex. H ]
7
.
30
1
Ans. − .
6
Ans. −
[ SM 7.6.13 ]
2.
[ Sh 9.1.6(b) ]
31. lim
x→0
tan x − x
x3
Ans.
1
.
3
[ SM 7.6.14 ]
Ans. Does not exist.
[ H 4.8.18(c) ]
4
32. lim
x→2
33. lim
1
Ans. .
π
x→3π
Ans. 0.
[ M 62.32 ]
1 + tan(x/4)
cos(x/2)
Ans.
32(1 − ln 2)
.
π
Ans. 1.
[ W VIII.1.2 ]
34. lim x1/(ln x)
Ans. e.
35. lim (cos x)csc x
Ans. 1.
x→∞
[ R 4.7.51 ]
1
Ans. − .
2
x
x −4
sin(πx)
Ans. 0.
[ W VIII.1.7 ]
18. lim sin x ln x
1
Ans. − .
2
−1
2
ex − e4
11. lim
x→2
x−2
√
3
x
12. lim
x→∞ ln x
ln(tan x)
13. lim
x→π/4 sin x − cos x
Ans. −1.
x→0
36.
lim
x→π/2−
cos x ln tan x
37. lim xsin(1/x)
x→∞
Ans. 0.
[ SSJ 124.17 ]
Ans. 1.
38.
h π
lim
2
x→π/2
i
− x tan x
Ans. 1.
[ SSJ 124.24 ]
sin 3x cot 2x
ln cos x
1
cot x
40. lim
−
x→0 x2
x
39.
Ans. −∞.
lim
x→0+
55. lim
x→∞
56.
lim
x→+∞
sin−1 x
x→0 x2 csc x
πx ln x
42. lim tan
x→1
2
Ans. 1.
2
Ans. − .
π
? 58. lim
x→0
43.
44.
x
xe − 4 + 2e − 2x
1 + x sin πx + x/2 + 2 sin πx
Ans. 2e−2 − 4.
√
lim− 1 − x ln ln(1/x)
Ans. 0.
lim
x→−2
x→1
[ W VIII.3.20 ]
−1
45. lim (csc x)(cos
Ans. e−π/2 .
x)/(ln x)
x→0+
2 sin θ − sin 2θ
46. lim
θ→0 sin θ − θ cos θ
47. lim
x→∞
48.
lim
θ→π/2
Ans. 3.
[ R Ch. 4 rev. 113 ]
ex + x
sinh x
Ans. 2.
2(ecos θ + θ − 1) − π
ln sin(−3θ)
2
Ans. − .
9
[ SSJ 124.9 ]
x sin(1/x)
ln x
x→0
x cot x
50. lim+ x
e −1
x→0
x ln x
51. lim
,
+
x→0 ln(1 + ax)
Ans. 0.
49. lim+
Ans. ∞.
a>0
Ans. −∞.
[ W VIII.1.10 ]
sin x
x
Ans. 0.
[ W VIII.1.5 ]
1/x2
Ans. e−1/6 .
[ R 4.7.75(a) ]
1
1
−
sin2 x x2
Ans.
1
.
3
[ R 4.7.75(b) ]
√
x ln x ln x x
√
√ ln x
x
(ln x) x
√
[ R 4.7.26 ]
x
x10
ex
? 57. lim+
x→0
41. lim
Ans. 1.
[ W VIII.4.4 ]
10
1
Ans. .
3
[ S 21.1(e) ]
tan−1 x − π/2 + x−1
coth−1 x − x−1
?? 59.
lim
x→+∞
60. Compute lim
x→0
x2 cos x
2 sin2 12 x
Ans. ∞.
[ W VIII.4.11 ]
by two methods.
Ans. 2.
[ M 62.10 ]
61. Finding the following limit was the first
example that L’Hôpital gave in demonstrating the rule that bears his name. Let
a be a positive constant and find
√
√
3
2a3 x − x4 − a a2 x
√
lim
.
4
x→a
a − ax3
Ans. 0.
[ Sh 9.1.16 ]
62. Show that L’Hôpital’s rule applies to the
x + cos x
, but that it is of no
limit lim
x→∞ x − cos x
help. Then evaluate the limit directly.
Ans. 1.
[ R 4.7.69 ]
Ans. 1.
? 63. Without using L’Hôpital’s rule, the limit
sin x
lim
can be evaluated by a rather
x→0
x
intricate geometric argument. Show that
it can be evaluated easily using L’Hôpital’s
rule. Then explain why doing so involves
circular reasoning.
[ R 4.7.72 ]
[H] Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum, et al. Calculus: Single and Multivariable, second edition. Wiley, 1998.
[M] Ross R. Middlemiss. Differential and Integral Calculus, first edition. McGraw-Hill, 1940.
[R] Jon Rogawski. Calculus: Early Transcendentals.
Freeman, 2008.
[S] Ivan S. Sokolnikoff. Advanced Calculus. McGrawHill, 1939.
[Sh] Shahriar Shahriari. Approximately Calculus. American Mathematical Society, 2006.
[SM] Robert T. Smith and Roland B. Minton. Calculus,
second edition. McGraw-Hill, 2002.
[SSJ] Edward S. Smith, Meyer Salkover, and Howard K.
Justice. Calculus. Wiley, 1938.
[W] David V. Widder. Advanced Calculus. PrenticeHall, 1947.
sin−1 x
52. lim
x→0 x cos−1 x
ln x
53. lim+
x→0 cot x
54. lim (ax)b/(cx) ,
x→∞
2
Ans. .
π
Ans. 0.
[ SM 7.6.31 ]
a, c 6= 0
References: