AP Calculus AB: 12.2.3 L'Hôpital's Rule and One to the Infinite Power
This content focuses on handling the indeterminate form 1 to the infinite power, which arises when a base approaching 1 is raised to an exponent growing without bound. To evaluate such limits, the expression is first rewritten using logarithmic identities.
L’Hôpital’s Rule and 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
L’Hôpital’s Rule and 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
You may encounter a limit that produces one to the infinite power, , which is another indeterminate form. It could be one, because one to a...
Evaluate limx →0+ (1+3x)^1/2x
e^ 3/2
Evaluate limx→0+ x^x
1
Evaluate lim x→∞ (1+5/x)^x
e^5
Evaluate limx→∞(1+1/x^2)x.
1
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| Term | Definition |
|---|---|
L’Hôpital’s Rule and One to the Infinite Power |
|
note |
|
Evaluate limx →0+ (1+3x)^1/2x | e^ 3/2 |
Evaluate limx→0+ x^x | 1 |
Evaluate lim x→∞ (1+5/x)^x | e^5 |
Evaluate limx→∞(1+1/x^2)x. | 1 |
Evaluate limx→0(1+2x+x^2)1/x | e^ 2 |