evaluate-each-limit-if-it-exists-use-l-hospitals-rule-if-appropriate-lim-_x-rightarrow-0-x-ln-sin-x

Question

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

Gregory Higby · Accepted Answer

we are evaluating the limits of uh X approaches zero of X times natural log of sine of X. Now, what I would do is rewrite this problem as the limit as X approaches zero of natural log of sinopec&#x27;s over one over X. Um Just because as you think of about um I don&#x27;t know if you need to think a lot, but we&#x27;re actually gonna get negative infinity over infinity. Um And the reason for that is because if you were to look at the first of all sign of zero is zero. That for you to think of natural log, it&#x27;s going infinitely down if I have zero from the right side. And then lastly thinking about one over X as a graph that I guess if we&#x27;re at the time I was here on the right side, we&#x27;re looking at that um Mhm. So from here we can actually evaluate the derivative of the top. So we have the limits as X goes to zero. Well, the drift of the top natural log is one over that piece. But then you have to multiply by the derivative of the inside, which would be co sign. And if you think about as one over X is the same thing as X to the negative first power. It&#x27;s a negative one X to the negative second power. Uh So we could rewrite this the limit as X goes to zero of negative X squared because remember we&#x27;re dividing by it so to move it back up to the numerator and then uh you know times co sign over sign. And now when we do direct substitution, what we end up with is negative of zero squared and co sign of zero is one over sign of 00 I should put you putting arrows in here again. So we end up with 0/0 again. So I guess we need to take the derivative of this again. So we&#x27;re going to have the limit as X goes to zero. But this time if I do the drift of the top, it&#x27;s going to be the um product rule. So the derivative of X squared is two X. Leave co Cinemax alone. Then now lee plus and you leave the negative X squared alone times the derivative of cosine, which means positive sign. Um All over the derivative of sine, which would be co signed. And now when we go to plug in we end up with negative two times zero times one plus zero times zero. All over one. Well we have all these zeros divided by one will just give me zero as the correct answer. So we had to do a lot to tell Israel twice

Carlos Pinilla · Answer

you have to believe it. Six approaches you. Um you&#x27;re right side. So I mean that, uh, exist. This positive ghost is hearing desertion off the potion. Could turns in the bags minus more for X also seems existed. It&#x27;s positive this part will, Will will also be positive. Mom. Rex Mine Is it being that&#x27;ll contended this sign called Coursing Over, so no thanks. Over sign so thes exhausted here will go to you&#x27;re also from Santo will be something for me. So you have to Oh, it believe these expression to get something more. Bill Murray. No sign of Becks sign. So you&#x27;re gonna have these corrections. Well, uh uh, What was that? So X tentacle St Main was one thing&#x27;s sign. X sign legs. My nose begs all that over over these two looks them saddlebags X that sandal big. So, uh, we have, ah picks. I&#x27;m just a final. Burke&#x27;s the lighting, so we&#x27;ll get this expression. Yeah. Oh, our sex hostess Here. Here you go to you know, im 00 Your time&#x27;s want is you. You&#x27;re minus zero. You have a form, Cyril over zero. Sure. Organ gonna play a little is exposure. So these women? 60 so Well, Well, they&#x27;re drove. These that is probably so off. Just one. So just we&#x27;ll just call. Sign affects fine ex bombs at the golf course signs minus Sign my sign of eggs on the earth of minus sign. He&#x27;s my school son. So you have This is living there. Functions up. I&#x27;m a born. Whoa! Can we have a bird Octo friendship The X just one. So one sign of bags I&#x27;m out plus vx constant on different shit. Sign differentiated. Signed You get out. We&#x27;ll sign of bags. I&#x27;m here. These people well believe you know six ghost zero these from there. Right inside. So, uh oh, yeah. This co sign will counsel with the sign Living as we egg My no sex I&#x27;m sign of bricks over a sign of books plus x school. Simon picks. So it&#x27;s excuses. You owe the ex multi from here. So that&#x27;s zero sign of 00 some balls on you. So again, how long? Zero over zero with these two councils. Those who are young. It&#x27;s what I want. You only have the extends sine minus that went up so well. We gonna play love pavilion. So the friendship this but it divided by the French. So, uh huh. So, uh, what? We don&#x27;t, uh beautiful fix. Minus six times I is gonna be Well, yeah. Wolf exes. One just It would be they&#x27;re Peru. That part of the protocol will be just minor side effects. I know you keep this one constant, but the science cool side. So much. Thanks. Now here, on bottom already. People sinus co sign here, then you have a product. So keep But I don&#x27;t know the friendship, this 1st 1 So just one times course goes, I&#x27;m now plus X keep this one called something different shape that the golf course only is my assignment Thames. So X times minus Sign of bricks so well here. We&#x27;re looking at the live read six approaches. Sirrah, you know, there are zero. So, uh well, here, uh, well, you have, uh, sign off zero zero zero times one, because you&#x27;ve got no sign of you. She&#x27;s one but zero up on here on bottom. Well, this is one they resolved one. Yeah, that is your soul off. That would be zero. My zero. We won. Great. So you can just forget it. That&#x27;s it&#x27;s here, Right? So you had yeah, to play little twice, but But you get the answer. It was not so much.

Carson Merrill · Answer

mhm. For this problem, we want to find the limit as X goes to zero. So um we start by taking the natural log of both sides. And in doing that we can use certain properties to get into a form that we want and that&#x27;s going to be the natural log of 1 -2. X over X. However, when we take the limit we end up getting 0/0, which is the indeterminate form. So we&#x27;re going to take the derivative of both the top and the bottom giving us um as a result will get the limit as X coke zero of one over 1 -2. X. Now, when we take the limit we end up getting negative two Because it&#x27;s negative two times the limit. So we end up getting just negative too. Remember that that is going to be The limit as X approaches zero of the natural log of wine. So as a result of that, we end up um evaluating this and using what battles rule and that will give us That E to the -2 will be our final answer.

Amrita Bhasin · Answer

Okay. The first thing we can do is we can rewrite this so it&#x27;s easier to manipulate. Won over. Sign of acts minus co signed X. We&#x27;re sign Axe. When we plug in, we simply end up with zero all over again. Therefore, we know we have to use the hoppy tolls. Rule the derivative of one issue zero the derivative of negative coast on Axis. Positive sign ups. The derivative of Sine X is co sign axe. Pretty straightforward the end of the zero for one, but this gives us a solution of zero.

Carlos Pinilla · Answer

until we help here. The living laser&#x27;s weeks. That was zero. But, uh, course he won&#x27;t fix miners. Who? Dungeon X also, uh, six roast this year. All this is one over flying uncle tendencies. No sign of that Or son, but, uh, exposed to zero here. Well, this you go to last year, community. Our last baby. It&#x27;s something like Is depending on which side you&#x27;re going Like you don&#x27;t know how to handle that, so Oh, we can manipulate this expression since we have a proportion. It&#x27;s just just, uh, so these course you can mine. It&#x27;s called dungeon. These photo we need X zero. Well, well, well. You know the same. You know where you&#x27;re home. Same denominator. So we can just that one minus cycle or the wedding by signs. So you can just simply, uh, So I would look at this Expression six hostess here. Oh, well, singles one. So one minus one of these zoo. I&#x27;m signed up. Zero ghost zero. You have a form zero or zero. So we&#x27;re in good shape. The right play. I love you, Bill. Your goal tells you that. Mmm. All these little people to believe it, baby. bottom. So they&#x27;re at the world for one issue. So it&#x27;s you. Oh, my nose. It is. Of course, Sign is minus sign. So my nose my nose sign affects on the radio off science, he&#x27;s course. So the well, then minus the minus Consul side effects You read about coursing. So, what we need here these thieves us before six. Who&#x27;s this year? Off A ll. These expression is gonna be just just scientifics Oracle sign. So six goes zero. There is no problem there. One I&#x27;m Dad goes through issue. I got one, uh, ready when we had to play the ones on that we got on our answer. Nothing. It is He will do suit.

Carson Merrill · Answer

we&#x27;re gonna be taking the limit as X goes to infinity of X to the one over X. Um And we can simplify this by um considering the fact that if we plug in infinity we&#x27;re going to get infinity to the power of zero. So since that&#x27;s the indeterminate form, we&#x27;re going to um take the natural log of both sides. When we do that, we&#x27;ll end up getting at the natural log of Y equals one over X times the natural log of X, which is just the natural log of acts over X. Then putting infinity, we get infinity over infinity. So we must Take the derivative of both sides giving us one over X Right to Buy one. Now when we plug in infinity we will get one over infinity which is zero divided by one, which is zero. So zero is not our answer. Instead what we have. Is that the natural log of why equals zero is our answer. So let&#x27;s solve for why we know that we can take E. To the power of both sides. Can I ask why equals E. To the power of zero? So based on this, we see that why is going to equal one? So one ends up being our final final answer. And this was through using a method by which we take the national log on both sides and then employ the hotel&#x27;s rule.

Problem

Evaluate each limit (if it exists). Use $L$ Hospi…

Problem 19 Easy Difficulty

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

Answer

0

Related Courses

Basic Technical Mathematics with Calculus

Related Topics

Discussion

Top Calculus 1 / AB Educators

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 27

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

0

Basic Technical Mathematics with Calculus

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus