12.2.4 Another Example of One to the Infinite Power

Question 1

Another Example of One to the Infinite Power

Answer 1

• Some indeterminate forms have to be transformed before you can apply L’Hôpital’s rule.
• In order to apply L’Hôpital’s rule to a limit of the form , use the properties of logarithms to rewrite the exponent as a logarithm.

Question 2

note

Answer 2

• When you encounter the indeterminate form , you will need to make use of two facts about exponents and logarithms.
• The first is that e raised to the natural log of any expression is equal to that same expression.
• The second is that when there is an exponent inside a natural log expression, it can be moved to the outside as a factor.
• Now that you have rewritten the expression, you can evaluate an easier limit. Forget about e and take the limit of its exponent.
• Remember that this sub-problem is not equal to the original limit. It is just a side calculation.
• To evaluate the limit in the sub-problem, you will have to transform the expression to produce an indeterminate
  quotient. Then you can apply L’Hôpital’s rule.
• The limit from the sub-problem is equal to –1, but that is not the value of the original limit!
• When you plug in the result of the side calculation, you get the value of the original limit.

Question 3

Evaluate limx→0+ x^tanx

Answer 3

1

Question 4

Evaluate limx→0+ (cotx)^sinx

Answer 4

1

Question 5

Evaluate limx→0 (1–x)^1/5x.

Answer 5

e^ −1/5

Question 6

Evaluate limx→2 (x^2)^1/ln(x–1).

Answer 6

√e

Question 7

Evaluate limx→0+ x^1/1+lnx.

Answer 7

e

Question 8

Evaluate limx→1+ (x – 1)^lnx.

Answer 8

1

Question 9

Evaluate limx→∞ (lnx)^1/x.

Answer 9

1

Question 10

Evaluate limx→0+ x^sinx.

Answer 10

1