Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Watch this step-by-step video, matched to your homework problem.

Try Numerade Free for 30 Days

Like

Report

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

0

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 7

L'Hospital's Rule

Derivatives

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

09:16

Find the limit. Use l'…

01:31

00:55

04:17

01:32

06:17

01:36

01:58

01:16

00:58

Determine each limit, if i…

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's over one over X. Um Just because as you think of about um I don't know if you need to think a lot, but we'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'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're at the time I was here on the right side, we'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'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'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're going to have the limit as X goes to zero. But this time if I do the drift of the top, it'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

View More Answers From This Book

Find Another Textbook

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

Find the limit. Use l'Hospital's Rule where appropriate. If there …

Determine each limit, if it exists.$$\lim _{x \rightarrow 0}(x \cot x)$$…

03:21

Give the proper trigonometric substitution and find the transformed integral…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.