L'Hopital's rule can be used to evaluate limits of indeterminate forms, such as 0/0, ∞/∞, and ∞*0. It states that if the limit of the numerator and denominator are both 0 (or ∞), the limit can be evaluated by taking the derivative of the numerator and denominator and evaluating the limit of their quotient. The rule effectively passes the calculation to the derivatives. Several examples are provided to demonstrate applying L'Hopital's rule to evaluate limits of various indeterminate forms.

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)
=