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Find the limits.$$\lim _{x \rightarrow+\infty}(\ln x)^{1 / x}$$

1

Calculus 1 / AB

Chapter 3

TOPICS IN DIFFERENTIATION

Section 6

L Hopital's Rule; Indeterminate Forms

Functions

Limits

Derivatives

Differentiation

Continuous Functions

Applications of the Derivative

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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So we're looking at the limit as X goes to it. Positive infinity of natural log of X raised to the one over X. Let's set that equal toe l Now let's take the limit. Excuse me. The natural log of each side. Okay. And our next step is to interchange the limit with the natural log. And we can do this because natural log. It's continuous. Okay, let's go ahead and use our log property to bring that one that one over X down. So now we have natural log of natural. Lagerback's divided by X. And if we try to evaluate this limit, we get infinity over infinity. So we're set up to use low petals rule. Let's go ahead and do that. So the derivative, the top is one over. Natural log X times one over X by chain room to remove. The bottom is just one. Okay. And this limit is something that we can easily evaluate. Zero. Okay, so we have that The natural log of LZ zero. This implies that l is equal to one. That's it. We're done

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