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.)
ex − x − 1 1
1. lim Ans. −1. 19. lim x sin Ans. 0.
x→0 cos x − 1 x→0 x
[ R 4.7 Ex. 4 ]
tan 3x 5
x3 − x2 − 10x − 8 3 20. lim Ans. .
2. lim Ans. . x→π/2 tan 5x 3
x→−2 5x3 + 12x2 − 2x − 12 5 [ R 4.7.18 ]
x−a sec x
3. lim Ans. a. 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. . 22. lim csc x − Ans. 0.
θ→0 θ2 2 x→0 x
[ M 62.12 ]
5. lim (sinh x − cosh x) Ans. 0. 23. lim (tan x)sin 2x Ans. 1.
x→π/2 [ S 21.3(b) ]
x→∞
tan x 24. lim x−3 ex Ans. 0.
6. lim + Ans. ∞. x→−∞
x→π/2 ln(2x − π) [ W VIII.2.1 ]
ln(t + 2)
tanh x 2 25. lim Ans. ln 2.
7. lim Ans. . x→∞ log2 t [ R Ch. 7 Rev. 115 ]
x→∞ tan−1 x π
sin x 1 − ln x
8. lim Ans. 1. 26. lim Ans. −1.
x→0 sinh x x/e − 1
x→e
x √ √
e 1 1 1 − tan x − 1 + tan x 1
9. lim − Ans. . 27. lim Ans. − .
x→0 ex − 1 x 2 x→π sin 2x 2
[ R Ch. 4 rev. 116 ] [ Sh 9.1.14 ]
ln(x − π/2) (π/2) − tan x −1
10. lim Ans. 0. 28. lim Ans. 1.
x→π/2 sec x [ SSJ 122 Ex. 4 ] x→∞ x−1 [ W VIII.1.3 Ex. H ]
2 √
ex − e4 x + 10 + 3x1/3 7
11. lim Ans. 4e4 . 29. lim Ans. − .
x→2 x−2 [ R 4.7.22 ] x→−1 4x2 + 3x − 1 30
√ sin x − x 1
3
x 30. lim Ans. − .
12. lim Ans. ∞. x→0 x3 6
x→∞ ln x
[ SM 7.6.13 ]
ln(tan x) √
13. lim Ans. 2. tan x − x 1
x→π/4 sin x − cos x 31. lim Ans. .
[ Sh 9.1.6(b) ] x→0 x3 3
[ SM 7.6.14 ]
e−x
14. lim Ans. Does not exist. 4 x
x→∞ sin x [ H 4.8.18(c) ]
x −4 32(1 − ln 2)
32. lim Ans. .
x→2 sin(πx) π
e−x
15. lim Ans. 0. 1 + tan(x/4)
x→∞ sin x + 2 33. lim Ans. 1.
x→3π cos(x/2)
tan−1 x − π4 1 [ W VIII.1.2 ]
16. lim Ans. .
x→1 tan π x − 1 π 34. lim x1/(ln x) Ans. e.
4
[ R 4.7.51 ] x→∞
tan−1 x 1 35. lim (cos x)csc x Ans. 1.
x→0
17. lim Ans. − .
x→−∞ cot−1 x 2 36. lim cos x ln tan x Ans. 0.
[ W VIII.1.7 ] x→π/2− [ SSJ 124.17 ]
18. lim sin x ln x Ans. 0.
x→0
[ M 62.32 ]
37. lim xsin(1/x) Ans. 1.
x→∞
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.)
ex − x − 1 1
1. lim Ans. −1. 19. lim x sin Ans. 0.
x→0 cos x − 1 x→0 x
[ R 4.7 Ex. 4 ]
tan 3x 5
x3 − x2 − 10x − 8 3 20. lim Ans. .
2. lim Ans. . x→π/2 tan 5x 3
x→−2 5x3 + 12x2 − 2x − 12 5 [ R 4.7.18 ]
x−a sec x
3. lim Ans. a. 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. . 22. lim csc x − Ans. 0.
θ→0 θ2 2 x→0 x
[ M 62.12 ]
5. lim (sinh x − cosh x) Ans. 0. 23. lim (tan x)sin 2x Ans. 1.
x→π/2 [ S 21.3(b) ]
x→∞
tan x 24. lim x−3 ex Ans. 0.
6. lim + Ans. ∞. x→−∞
x→π/2 ln(2x − π) [ W VIII.2.1 ]
ln(t + 2)
tanh x 2 25. lim Ans. ln 2.
7. lim Ans. . x→∞ log2 t [ R Ch. 7 Rev. 115 ]
x→∞ tan−1 x π
sin x 1 − ln x
8. lim Ans. 1. 26. lim Ans. −1.
x→0 sinh x x/e − 1
x→e
x √ √
e 1 1 1 − tan x − 1 + tan x 1
9. lim − Ans. . 27. lim Ans. − .
x→0 ex − 1 x 2 x→π sin 2x 2
[ R Ch. 4 rev. 116 ] [ Sh 9.1.14 ]
ln(x − π/2) (π/2) − tan x −1
10. lim Ans. 0. 28. lim Ans. 1.
x→π/2 sec x [ SSJ 122 Ex. 4 ] x→∞ x−1 [ W VIII.1.3 Ex. H ]
2 √
ex − e4 x + 10 + 3x1/3 7
11. lim Ans. 4e4 . 29. lim Ans. − .
x→2 x−2 [ R 4.7.22 ] x→−1 4x2 + 3x − 1 30
√ sin x − x 1
3
x 30. lim Ans. − .
12. lim Ans. ∞. x→0 x3 6
x→∞ ln x
[ SM 7.6.13 ]
ln(tan x) √
13. lim Ans. 2. tan x − x 1
x→π/4 sin x − cos x 31. lim Ans. .
[ Sh 9.1.6(b) ] x→0 x3 3
[ SM 7.6.14 ]
e−x
14. lim Ans. Does not exist. 4 x
x→∞ sin x [ H 4.8.18(c) ]
x −4 32(1 − ln 2)
32. lim Ans. .
x→2 sin(πx) π
e−x
15. lim Ans. 0. 1 + tan(x/4)
x→∞ sin x + 2 33. lim Ans. 1.
x→3π cos(x/2)
tan−1 x − π4 1 [ W VIII.1.2 ]
16. lim Ans. .
x→1 tan π x − 1 π 34. lim x1/(ln x) Ans. e.
4
[ R 4.7.51 ] x→∞
tan−1 x 1 35. lim (cos x)csc x Ans. 1.
x→0
17. lim Ans. − .
x→−∞ cot−1 x 2 36. lim cos x ln tan x Ans. 0.
[ W VIII.1.7 ] x→π/2− [ SSJ 124.17 ]
18. lim sin x ln x Ans. 0.
x→0
[ M 62.32 ]
37. lim xsin(1/x) Ans. 1.
x→∞
