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Mathematics /AP Calculus AB: 12.2.3 L'Hôpital's Rule and One to the Infinite Power

AP Calculus AB: 12.2.3 L'Hôpital's Rule and One to the Infinite Power

Mathematics7 CardsCreated 7 months ago

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.

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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.

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Key Terms

Term

Definition

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

Evaluate limx→0(1+2x+x^2)1/x

e^ 2

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AP Calculus AB: 12.2.3 L'Hôpital's Rule and One to the Infinite Power

Term	Definition
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.
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 any power is one. Or it could be infinity, because it began as one and a tiny bit more, which grows large when raised to infinity. If you encounter a limit that produces this form, you will need to transform the expression. The key is to raise the number e to the natural log of the expression. This equals the original expression. Once you have transformed the original limit, you can focus on the expression to which e is raised. This new limit does not equal the original limit. It is a sub-problem. It produces an indeterminate product, which you must transform into an indeterminate quotient. Now you can use L’Hôpital’s rule. The sub-problem limit equals one. By plugging in the value of the limit into the sub-problem, you can evaluate the original limit. Since e raised to the first power is still e, that’s your answer. Some mathematicians use this limit expression as an alternate definition for e.
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