Problem

Find the limits. $$ \lim _{x \rightarrow \pi / 2^…

01:05

Problem 24 Medium Difficulty

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

Answer

$$\text { The limit is equal to } 0 .$$

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Calculus 1 / AB

Calculus Early Transcendentals

Chapter 3

TOPICS IN DIFFERENTIATION

Section 6

L Hopital's Rule; Indeterminate Forms

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Video Transcript

were considering the limits as X goes to zero from the positive side of 10 X times Natural log X. Let's go ahead and rewrite this. We have limits as X goes to zero from the positive side. Natural log on top, kusi can't on the bottom and one over co sign. So we just use the definition of tangent and change the sign to a one over Cosi can't. Now let's go ahead and use one of our limit properties that says that the limit of a product is equal to the product of limit. That is, we're going to split up this into two limits. And we're doing this because we know what the limit of one over co sign as experts. Zero is OK, but let's not go ahead and evaluate yet. So we see that this first limit here we he should be able to play low petals rule to because as the the numerator as X approaches here from the positive side, the numerator is approaching negative infinity and so is the denominator. Excuse me, The denominator is actually approaching positive infinity, where the numerator is approaching negativity. We still been in attributed form, though Okay, so let's go ahead and use low Patel's rule. So we have one over X on the top, and we have negative Cosi can't x times co tangent on the bottom. And then we have this other limit thing here. Okay, um, let's go ahead and rewrite that. And now, uh, this limit we see is one. So I'm going to go ahead and evaluate that, not read it in this next step. So we have negative sine x over X Times tangent, and that's what we're left with. Let's go ahead and split this up into two limits. Okay, so we see that this first limit, we can apply low battles rule to because it as an indeterminant form of 0/0. So let's go ahead and do that. And the second limits is something that we can evaluate. So Lou petals rule. We get negative co sign acts over one. Then we have this LTD's X ghost zero of 10 x. Okay, so we've negative one times zero, This is zero. So that's our answer. Zero, and that's it.

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