L'Hopital's Rule

1. L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

2. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then
then lim f(x)
g(x)
= lim f '(x)
g'(x)
if the limit exists, or its ±∞ .
xa xa
xa xa
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

3. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then
then lim f(x)
g(x)
= lim f '(x)
g'(x)
if the limit exists, or its ±∞ .
xa xa
xa xa
We call this type of limit as the " 0 "
0
indeterminate
type.
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

4. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then
then lim f(x)
g(x)
= lim f '(x)
g'(x)
if the limit exists, or its ±∞ .
xa xa
xa xa
We call this type of limit as the " 0 "
0
indeterminate
L'Hopital's rule passes the calculation of the
" 0 "
0
form to the calculation of derivatives.
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
type.

5. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then
then lim f(x)
g(x)
= lim f '(x)
g'(x)
if the limit exists, or its ±∞ .
xa xa
xa xa
We call this type of limit as the " 0 "
0
indeterminate
L'Hopital's rule passes the calculation of the
" 0 "
0
form to the calculation of derivatives.
Example: A. Find lim
sin(x)
xx0
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
type.

6. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then
then lim f(x)
g(x)
= lim f '(x)
g'(x)
if the limit exists, or its ±∞ .
xa xa
xa xa
We call this type of limit as the " 0 "
0
indeterminate
L'Hopital's rule passes the calculation of the
" 0 "
0
form to the calculation of derivatives.
Example: A. Find lim
sin(x)
xx0
Since lim sin(x) = 0 and lim x = 0,
x0 x0
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
type.

7. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then
then lim f(x)
g(x)
= lim f '(x)
g'(x)
if the limit exists, or its ±∞ .
xa xa
xa xa
We call this type of limit as the " 0 "
0
indeterminate
L'Hopital's rule passes the calculation of the
" 0 "
0
form to the calculation of derivatives.
Example: A. Find lim
sin(x)
xx0
Since lim sin(x) = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
type.

8. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then
then lim f(x)
g(x)
= lim f '(x)
g'(x)
if the limit exists, or its ±∞ .
xa xa
xa xa
We call this type of limit as the " 0 "
0
indeterminate
L'Hopital's rule passes the calculation of the
" 0 "
0
form to the calculation of derivatives.
Example: A. Find lim
sin(x)
xx0
Since lim sin(x) = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
lim
sin(x)
x = lim
[sin(x)]'
[x]'x0 x0
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
type.

9. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then
then lim f(x)
g(x)
= lim f '(x)
g'(x)
if the limit exists, or its ±∞ .
xa xa
xa xa
We call this type of limit as the " 0 "
0
indeterminate
L'Hopital's rule passes the calculation of the
" 0 "
0
form to the calculation of derivatives.
Example: A. Find lim
sin(x)
xx0
Since lim sin(x) = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
lim
sin(x)
x = lim
[sin(x)]'
[x]'x0 x0
= lim
cos(x)
1x0
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
type.

10. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then
then lim f(x)
g(x)
= lim f '(x)
g'(x)
if the limit exists, or its ±∞ .
xa xa
xa xa
We call this type of limit as the " 0 "
0
indeterminate
L'Hopital's rule passes the calculation of the
" 0 "
0
form to the calculation of derivatives.
Example: A. Find lim
sin(x)
xx0
Since lim sin(x) = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
lim
sin(x)
x = lim
[sin(x)]'
[x]'x0 x0
= lim
cos(x)
1x0
= 1.
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
type.

11. B. Find lim ex – 1
xx0
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

12. B. Find lim ex – 1
xx0
Since lim ex – 1 = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

13. B. Find lim ex – 1
xx0
Since lim ex – 1 = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
lim ex – 1
x = lim [ex – 1 ]'
[x]'x0 x0
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

14. B. Find lim ex – 1
xx0
Since lim ex – 1 = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
lim ex – 1
x = lim [ex – 1 ]'
[x]'x0 x0
= lim
ex
1x0
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

15. B. Find lim ex – 1
xx0
Since lim ex – 1 = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
lim ex – 1
x = lim [ex – 1 ]'
[x]'x0 x0
= lim
ex
1x0
= 1.
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

16. B. Find lim ex – 1
xx0
Since lim ex – 1 = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
lim ex – 1
x = lim [ex – 1 ]'
[x]'x0 x0
= lim
ex
1x0
= 1.
C. Find lim ex – 1
x2x0+
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

17. B. Find lim ex – 1
xx0
Since lim ex – 1 = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
lim ex – 1
x = lim [ex – 1 ]'
[x]'x0 x0
= lim
ex
1x0
= 1.
C. Find lim ex – 1
x2x0+
Since lim ex – 1 = 0 and lim x2 = 0, use the L'Hopital's
rule:
x0+ x0+
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

18. B. Find lim ex – 1
xx0
Since lim ex – 1 = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
lim ex – 1
x = lim [ex – 1 ]'
[x]'x0 x0
= lim
ex
1x0
= 1.
C. Find lim ex – 1
x2x0+
Since lim ex – 1 = 0 and lim x2 = 0, use the L'Hopital's
rule: lim ex – 1
x2 = lim [ex – 1 ]'
[x2]'x0+
x0+ x0+
x0+
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

19. B. Find lim ex – 1
xx0
Since lim ex – 1 = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
lim ex – 1
x = lim [ex – 1 ]'
[x]'x0 x0
= lim
ex
1x0
= 1.
C. Find lim ex – 1
x2x0+
Since lim ex – 1 = 0 and lim x2 = 0, use the L'Hopital's
rule: lim ex – 1
x2 = lim [ex – 1 ]'
[x2]'x0+
= lim
ex
2xx0+
x0+ x0+
x0+
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

20. B. Find lim ex – 1
xx0
Since lim ex – 1 = 0 and lim x = 0, use the L'Hopital's
rule:
x0 x0
lim ex – 1
x = lim [ex – 1 ]'
[x]'x0 x0
= lim
ex
1x0
= 1.
C. Find lim ex – 1
x2x0+
Since lim ex – 1 = 0 and lim x2 = 0, use the L'Hopital's
rule: lim ex – 1
x2 = lim [ex – 1 ]'
[x2]'x0+
= lim
ex
2xx0+
= ∞.
x0+ x0+
x0+
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

21. D. Find lim 1
x(sin(1/x2))x+∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

22. D. Find lim 1
x(sin(1/x2))x+∞
Put the limit in the form of .
1/x
sin(1/x2)
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

23. D. Find lim 1
x(sin(1/x2))x+∞
Put the limit in the form of .
lim
1/x
x+∞
1/x = lim
x+∞
sin(1/x2) = 0, use the L'Hopital's rule:
sin(1/x2)
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

24. D. Find lim 1
x(sin(1/x2))x+∞
Put the limit in the form of .
lim
1/x
x+∞
1/x = lim
x+∞
sin(1/x2) = 0, use the L'Hopital's rule:
lim 1
x(sin(1/x2))x+∞
sin(1/x2)
= lim 1/x
sin(1/x2)x+∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

25. D. Find lim 1
x(sin(1/x2))x+∞
Put the limit in the form of .
lim
1/x
x+∞
1/x = lim
x+∞
sin(1/x2) = 0, use the L'Hopital's rule:
lim 1
x(sin(1/x2))x+∞
sin(1/x2)
= lim 1/x
sin(1/x2)x+∞
= lim
[1/x]'
[sin(1/x2)]'x+∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

26. D. Find lim 1
x(sin(1/x2))x+∞
Put the limit in the form of .
lim
1/x
x+∞
1/x = lim
x+∞
sin(1/x2) = 0, use the L'Hopital's rule:
lim 1
x(sin(1/x2))x+∞
sin(1/x2)
= lim 1/x
sin(1/x2)x+∞
= lim
[1/x]'
[sin(1/x2)]'x+∞
= lim
-x-2
cos(1/x2)(-2x-3)x+∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

27. D. Find lim 1
x(sin(1/x2))x+∞
Put the limit in the form of .
lim
1/x
x+∞
1/x = lim
x+∞
sin(1/x2) = 0, use the L'Hopital's rule:
lim 1
x(sin(1/x2))x+∞
sin(1/x2)
= lim 1/x
sin(1/x2)x+∞
= lim
[1/x]'
[sin(1/x2)]'x+∞
= lim
-x-2
cos(1/x2)(-2x-3)x+∞
= lim x
2cos(1/x2)x+∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

28. D. Find lim 1
x(sin(1/x2))x+∞
Put the limit in the form of .
lim
1/x
x+∞
1/x = lim
x+∞
sin(1/x2) = 0, use the L'Hopital's rule:
lim 1
x(sin(1/x2))x+∞
sin(1/x2)
= lim 1/x
sin(1/x2)x+∞
= lim
[1/x]'
[sin(1/x2)]'x+∞
= lim
-x-2
cos(1/x2)(-2x-3)x+∞
= lim x
2cos(1/x2)x+∞
= ∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

29. Remark: We need lim f(x) = lim g(x) = 0 for
L'Hopital's Rule (for the 0/0-form).
xa xa
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

30. Remark: We need lim f(x) = lim g(x) = 0 for
L'Hopital's Rule (for the 0/0-form).
xa xa
E. lim 2x
3x + 2x0
Example:
= 0
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

31. Remark: We need lim f(x) = lim g(x) = 0 for
L'Hopital's Rule (for the 0/0-form).
xa xa
E. lim 2x
3x + 2x0
Example:
= 0 = lim [2x]'
[3x + 2]'x0
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

32. Remark: We need lim f(x) = lim g(x) = 0 for
L'Hopital's Rule (for the 0/0-form).
xa xa
E. lim 2x
3x + 2x0
Example:
= 0 = lim [2x]'
[3x + 2]'x0 = 2
3 .
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

33. Remark: We need lim f(x) = lim g(x) = 0 for
L'Hopital's Rule (for the 0/0-form).
xa xa
E. lim 2x
3x + 2x0
Example:
= 0 = lim [2x]'
[3x + 2]'x0 = 2
3 .
are of the
" ∞ " indeterminate forms.∞
If lim f(x) = lim g(x) = ∞, then limxa xa
f(x)
g(x)xa
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

34. Remark: We need lim f(x) = lim g(x) = 0 for
L'Hopital's Rule (for the 0/0-form).
xa xa
E. lim 2x
3x + 2x0
Example:
= 0 = lim [2x]'
[3x + 2]'x0 = 2
3 .
are of the
" ∞ " indeterminate forms. L'Hopital’s Rule applies∞
If lim f(x) = lim g(x) = ∞, then limxa xa
f(x)
g(x)xa
in these situations also,
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

35. Remark: We need lim f(x) = lim g(x) = 0 for
L'Hopital's Rule (for the 0/0-form).
xa xa
E. lim 2x
3x + 2x0
Example:
= 0 = lim [2x]'
[3x + 2]'x0 = 2
3 .
are of the
" ∞ " indeterminate forms. L'Hopital’s Rule applies∞
If lim f(x) = lim g(x) = ∞, then limxa xa
f(x)
g(x)xa
in these situations also,
lim f(x)
g(x)
= lim f '(x)
g'(x).xa xa
that is
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

36. Example: F. Find lim
3x2 – 4
2x2 + x – 5x∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

37. Example: F. Find lim
3x2 – 4
2x2 + x – 5x∞
It’s the form." ∞ "
∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

38. Example: F. Find lim
3x2 – 4
2x2 + x – 5x∞
It’s the form." ∞ "
∞
Hence 3x2 – 4
2x2 + x – 5
limx∞
=
[3x2 – 4]'
[2x2 + x – 5]'
limx∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

39. Example: F. Find lim
3x2 – 4
2x2 + x – 5x∞
It’s the form." ∞ "
∞
Hence 3x2 – 4
2x2 + x – 5
limx∞
=
[3x2 – 4]'
[2x2 + x – 5]'
limx∞
6x
4x + 1
limx∞
=
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

40. Example: F. Find lim
3x2 – 4
2x2 + x – 5x∞
It’s the form." ∞ "
∞
Hence 3x2 – 4
2x2 + x – 5
limx∞
=
[3x2 – 4]'
[2x2 + x – 5]'
limx∞
6x
4x + 1
limx∞
= use L'Hopital's Rule again
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

41. Example: F. Find lim
3x2 – 4
2x2 + x – 5x∞
It’s the form." ∞ "
∞
Hence 3x2 – 4
2x2 + x – 5
limx∞
=
[3x2 – 4]'
[2x2 + x – 5]'
limx∞
6x
4x + 1
limx∞
=
= use L'Hopital's Rule again
[6x]'
[4x + 1]'
limx∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

42. Example: F. Find lim
3x2 – 4
2x2 + x – 5x∞
It’s the form." ∞ "
∞
Hence 3x2 – 4
2x2 + x – 5
limx∞
=
[3x2 – 4]'
[2x2 + x – 5]'
limx∞
6x
4x + 1
limx∞
=
= use L'Hopital's Rule again
[6x]'
[4x + 1]'
limx∞
= 6
4
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
= 3
2

43. G. ex
P(x)x∞
where p(x) is a polynomial.Find lim
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

44. G. ex
P(x)x∞
where p(x) is a polynomialFind lim
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
of degree > 0.

45. It’s the form." ∞ "
∞
Hence limx∞ = limx∞
ex
P(x)
[ex]'
[P(x)]'
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
G. ex
P(x)x∞
where p(x) is a polynomialFind lim
of degree > 0.

46. It’s the form." ∞ "
∞
Hence limx∞ = limx∞
ex
P(x)
[ex]'
[P(x)]'
= limx∞
ex
[P(x)]'
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
G. ex
P(x)x∞
where p(x) is a polynomialFind lim
of degree > 0.

47. It’s the form." ∞ "
∞
Hence limx∞ = limx∞
ex
P(x)
[ex]'
[P(x)]'
= limx∞
ex
[P(x)]'
Use the L'Hopital's Rule repeatedly if necessary,
eventually the denominator is a non-zero constant K.
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
G. ex
P(x)x∞
where p(x) is a polynomialFind lim
of degree > 0.

48. It’s the form." ∞ "
∞
Hence limx∞ = limx∞
= limx∞
ex
K
ex
P(x)
[ex]'
[P(x)]'
= limx∞
ex
[P(x)]'
Use the L'Hopital's Rule repeatedly if necessary,
eventually the denominator is a non-zero constant K.
Hence limx∞
ex
P(x)
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
G. ex
P(x)x∞
where p(x) is a polynomialFind lim
of degree > 0.

49. It’s the form." ∞ "
∞
Hence limx∞ = limx∞
= limx∞
ex
K
= ∞
ex
P(x)
[ex]'
[P(x)]'
= limx∞
ex
[P(x)]'
Use the L'Hopital's Rule repeatedly if necessary,
eventually the denominator is a non-zero constant K.
Hence limx∞
ex
P(x)
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
G. ex
P(x)x∞
where p(x) is a polynomialFind lim
of degree > 0.

50. It’s the form." ∞ "
∞
Hence limx∞ = limx∞
= limx∞
ex
K
We say that ex goes to ∞ faster than any polynomial.
= ∞
ex
P(x)
[ex]'
[P(x)]'
= limx∞
ex
[P(x)]'
Use the L'Hopital's Rule repeatedly if necessary,
eventually the denominator is a non-zero constant K.
Hence limx∞
ex
P(x)
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
G. ex
P(x)x∞
where p(x) is a polynomialFind lim
of degree > 0.

51. H. xp
Ln(x)x∞
where p > 0.Find lim
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

52. H. xp
Ln(x)x∞
It’s the form." ∞ "
∞
where p > 0.Find lim
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

53. H. xp
Ln(x)x∞
It’s the form." ∞ "
∞
Hence limx∞ =
where p > 0.Find lim
limx∞
xp
Ln(x)
[xp]'
[Ln(x)]'
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

54. H. xp
Ln(x)x∞
It’s the form." ∞ "
∞
Hence limx∞ =
where p > 0.Find lim
limx∞
= limx∞
pxp-1
x-1
xp
Ln(x)
[xp]'
[Ln(x)]'
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

55. H. xp
Ln(x)x∞
It’s the form." ∞ "
∞
Hence limx∞ =
where p > 0.Find lim
limx∞
= limx∞
pxp-1
x-1
xp
Ln(x)
[xp]'
[Ln(x)]'
= limx∞
pxp
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

56. H. xp
Ln(x)x∞
It’s the form." ∞ "
∞
Hence limx∞ =
where p > 0.Find lim
limx∞
= limx∞
pxp-1
x-1
xp
Ln(x)
[xp]'
[Ln(x)]'
= limx∞
pxp = ∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

57. H. xp
Ln(x)x∞
It’s the form." ∞ "
∞
Hence limx∞ =
where p > 0.Find lim
limx∞
= limx∞
pxp-1
x-1
We say that xp (p > 0) goes to ∞ faster than Ln(x).
xp
Ln(x)
[xp]'
[Ln(x)]'
= limx∞
pxp = ∞
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

58. H. xp
Ln(x)x∞
It’s the form." ∞ "
∞
Hence limx∞ =
where p > 0.Find lim
limx∞
= limx∞
pxp-1
x-1
We say that xp (p > 0) goes to ∞ faster than Ln(x).
xp
Ln(x)
[xp]'
[Ln(x)]'
= limx∞
pxp = ∞
The ∞/∞ form is the same as ∞*0 form since
∞/∞ = ∞ * (1/∞) and we may view that 1/∞ as 0.
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

59. I.
x0+
Find lim sin(x)Ln(x)
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

60. I.
x0+
Find lim sin(x)Ln(x)
Lim sin(x) = 0 and lim Ln(x) = -∞ so its the 0*∞ form.
x0+ x0+
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

61. I.
x0+
Find lim sin(x)Ln(x)
Lim sin(x) = 0 and lim Ln(x) = -∞ so its the 0*∞ form.
x0+ x0+
Write it as
1/sin(x)
Ln(x)
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

62. I.
x0+
in the form.∞
∞
Find lim sin(x)Ln(x)
Lim sin(x) = 0 and lim Ln(x) = -∞ so its the 0*∞ form.
x0+ x0+
Write it as
csc(x)
Ln(x)
1/sin(x)
Ln(x)
=
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

63. I.
x0+
in the form.∞
∞
Hence lim sin(x)Ln(x) = lim
Find lim sin(x)Ln(x)
Lim sin(x) = 0 and lim Ln(x) = -∞ so its the 0*∞ form.
x0+ x0+
Write it as
csc(x)
Ln(x)
csc(x)
Ln(x)
x0
+
x0
+
1/sin(x)
Ln(x)
=
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

64. I.
x0+
in the form.∞
∞
Hence lim sin(x)Ln(x) = lim
=
Find lim sin(x)Ln(x)
lim
Lim sin(x) = 0 and lim Ln(x) = -∞ so its the 0*∞ form.
x0+ x0+
Write it as
csc(x)
Ln(x)
csc(x)
Ln(x)
x0
+
x0
+
[csc(x)]'
[Ln(x)]'
x0
+
=
1/sin(x)
Ln(x)
=
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

65. I.
x0+
in the form.∞
∞
Hence lim sin(x)Ln(x) = lim
=
Find lim sin(x)Ln(x)
lim
Lim sin(x) = 0 and lim Ln(x) = -∞ so its the 0*∞ form.
x0+ x0+
Write it as
csc(x)
Ln(x)
csc(x)
Ln(x)
x0
+
x0
+
[csc(x)]'
[Ln(x)]'
x0
+
= lim
-csc(x)cot(x)
1/x
x0
+
1/sin(x)
Ln(x)
=
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

66. I.
x0+
in the form.∞
∞
Hence lim sin(x)Ln(x) = lim
=
Find lim sin(x)Ln(x)
lim
Lim sin(x) = 0 and lim Ln(x) = -∞ so its the 0*∞ form.
x0+ x0+
Write it as
csc(x)
Ln(x)
csc(x)
Ln(x)
x0
+
x0
+
[csc(x)]'
[Ln(x)]'
x0
+
= lim
-csc(x)cot(x)
1/x
x0
+
= lim
x
- sin(x)tan(x)
x0
+
1/sin(x)
Ln(x)
=
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

67. I.
x0+
in the form.∞
∞
Hence lim sin(x)Ln(x) = lim
=
Find lim sin(x)Ln(x)
lim
Lim sin(x) = 0 and lim Ln(x) = -∞ so its the 0*∞ form.
x0+ x0+
Write it as
csc(x)
Ln(x)
csc(x)
Ln(x)
x0
+
x0
+
[csc(x)]'
[Ln(x)]'
x0
+
= lim
-csc(x)cot(x)
1/x
x0
+
= lim
x
- sin(x)tan(x)
x0
+
=
x
-sin(x)lim
x0
+
tan(x)lim
x0
+
1/sin(x)
Ln(x)
=
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

68. I.
x0+
in the form.∞
∞
Hence lim sin(x)Ln(x) = lim
=
Find lim sin(x)Ln(x)
lim
Lim sin(x) = 0 and lim Ln(x) = -∞ so its the 0*∞ form.
x0+ x0+
Write it as
csc(x)
Ln(x)
csc(x)
Ln(x)
x0
+
x0
+
[csc(x)]'
[Ln(x)]'
x0
+
= lim
-csc(x)cot(x)
1/x
x0
+
= lim
x
- sin(x)tan(x)
x0
+
=
x
-sin(x)lim
x0
+
tan(x)lim
x0
+
1
1/sin(x)
Ln(x)
=
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)

69. I.
x0+
in the form.∞
∞
Hence lim sin(x)Ln(x) = lim
=
Find lim sin(x)Ln(x)
lim
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
Lim sin(x) = 0 and lim Ln(x) = -∞ so its the 0*∞ form.
x0+ x0+
Write it as
csc(x)
Ln(x)
csc(x)
Ln(x)
x0
+
x0
+
[csc(x)]'
[Ln(x)]'
x0
+
= lim
-csc(x)cot(x)
1/x
x0
+
= lim
x
- sin(x)tan(x)
x0
+
=
x
-sin(x)lim
x0
+
tan(x)lim
x0
+
0-1
1/sin(x)
Ln(x)
=

70. I.
x0+
in the form.∞
∞
Hence lim sin(x)Ln(x) = lim
=
Find lim sin(x)Ln(x)
lim
L'Hopital's Rule (0/0, ∞/∞, ∞*0 forms)
Lim sin(x) = 0 and lim Ln(x) = -∞ so its the 0*∞ form.
x0+ x0+
Write it as
csc(x)
Ln(x)
csc(x)
Ln(x)
x0
+
x0
+
[csc(x)]'
[Ln(x)]'
x0
+
= lim
-csc(x)cot(x)
1/x
x0
+
= lim
x
- sin(x)tan(x)
x0
+
=
x
-sin(x)lim
x0
+
tan(x) = 0lim
x0
+
0-1
1/sin(x)
Ln(x)
=