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Evaluate each limit (if it exists). Use $L$ Hospital's rule (if appropriate).$$\lim _{x \rightarrow 0^{+}} \frac{\ln x}{x^{-1}}$$

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Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 7

L'Hospital's Rule

Derivatives

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

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In mathematics, precalculu…

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Find the limit. Use l'…

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Limits Evaluate the follow…

we're evaluating the limits As X approaches zero on the right side of natural log of X over instead of writing. Um Excellent. Negative empowerment. Right one over X. Uh and it might be helpful to look at the graphs. So natural log of X looks like this. So if I were to evaluate zero on the right side, it is approaching negative infinity. If I were to look at the graph of one over X, it looks like this. Well on the right side, that's positive infinity. Um So this is definitely the indeterminate form of well, it is technically negative infinity over infinity. But we have that indeterminate form that tells us that we can use the lumpy tiles rules. So what do we do? Um is we take So that original problem, it's the same limit as X approaches zero on the right side and you just take the derivative of the top. So the drift of the natural log Becks is one of our X. And maybe that's why they left it as X to the negative one power. Because the directive of that would be negative one X to the negative second power. So just a reminder that that is the same thing. Um has one over X over negative one over X squared. And just a reminder kind of running out of room. That's the same thing, hopes of parental rights from the right side as in sub divided by a fraction. We can multiply by the reciprocal. So that's the same thing as well. One of those exits will cancel out because there's two on top, there's one on bottom. So we're looking at the limit as X approaches zero on the right side of just negative X. And we can do direct substitution now To get zero as our final answer.

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