12.2.4 Another Example of One to the Infinite Power Flashcards Preview

AP Calculus AB > 12.2.4 Another Example of One to the Infinite Power > Flashcards

Flashcards in 12.2.4 Another Example of One to the Infinite Power Deck (10)
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1
Q

Another Example of One to the Infinite Power

A
  • 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.
2
Q

note

A
  • 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.
3
Q

Evaluate limx→0+ x^tanx

A

1

4
Q

Evaluate limx→0+ (cotx)^sinx

A

1

5
Q

Evaluate limx→0 (1–x)^1/5x.

A

e^ −1/5

6
Q

Evaluate limx→2 (x^2)^1/ln(x–1).

A

√e

7
Q

Evaluate limx→0+ x^1/1+lnx.

A

e

8
Q

Evaluate limx→1+ (x – 1)^lnx.

A

1

9
Q

Evaluate limx→∞ (lnx)^1/x.

A

1

10
Q

Evaluate limx→0+ x^sinx.

A

1

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