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Find the limits.$$\lim _{x \rightarrow 0^{+}}(-\ln x)^{x}$$

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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

Baylor University

University of Nottingham

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We're starting with the limit as X goes to zero from the positive side of negative natural log X rays to the X. Let's go ahead and assign that to variable L And then we're gonna take the natural log of each side. And now, for our next step, we're going to switch the natural log and the limit. And we can do this because natural August continuous. Okay? And so let's go ahead and rewrite this thing as a fraction using the natural log properties so you can take that X and bring it down. Rewrite it as one over X. Okay, um, now, notice that the numerator does in fact make sense because, as zero has extends to zero from the positive side, natural law give extends to negative infinity. So we have the numerator attending to positive infinity on the top and the denominator tending to positivity on the bottom. So we're good to go to use Lou petals rule. Let's go ahead and do that one over a natural log X times negative one over X and then the denominator. We have negative one over X squared, and now we can simplify this. Um, and this works out to be negative. X over natural log X and this we can evaluate it. Zero Okay, so we have natural law. Go back and we have natural log of l is equal to zero. That implies that are limits. L is equal to one and we're done.

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