AP Calculus AB: 12.2.4 Another Example of One to the Infinite Power
This content explains how to evaluate limits that result in the indeterminate form 1∞. It demonstrates the use of exponential and logarithmic identities to transform the expression: rewrite it as limit of the exponent limit of the exponent, where the exponent involves a logarithm. The transformed expression is easier to work with and often results in an indeterminate quotient, allowing the use of L’Hôpital’s Rule. The solution to this intermediate limit is then substituted back into the exponential to get the final result.
Another Example of One to the Infinite Power
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.
Key Terms
Another Example of One to the Infinite Power
Some indeterminate forms have to be transformed before you can apply L’Hôpital’s rule.
In order to apply L’Hôpital’s rule to...
note
When you encounter the indeterminate form , you will need to make use of two facts about exponents and logarithms.
The first...
Evaluate limx→0+ x^tanx
1
Evaluate limx→0+ (cotx)^sinx
1
Evaluate limx→0 (1–x)^1/5x.
e^ −1/5
Evaluate limx→2 (x^2)^1/ln(x–1).
√e
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| Term | Definition |
|---|---|
Another Example of One to the Infinite Power |
|
note |
|
Evaluate limx→0+ x^tanx | 1 |
Evaluate limx→0+ (cotx)^sinx | 1 |
Evaluate limx→0 (1–x)^1/5x. | e^ −1/5 |
Evaluate limx→2 (x^2)^1/ln(x–1). | √e |
Evaluate limx→0+ x^1/1+lnx. | e |
Evaluate limx→1+ (x – 1)^lnx. | 1 |
Evaluate limx→∞ (lnx)^1/x. | 1 |
Evaluate limx→0+ x^sinx. | 1 |