find-the-limits-lim-_x-rightarrow-0-6-x2cot-xcsc-2-x-2

Question

Find the limits. 
$$\lim _{x \rightarrow 0} 6 x^{2}(\cot x)(\csc 2 x)$$

Issa Dababneh · Accepted Answer

All right, so in this video were asked to find them. Limit is exposed to 06 x squared times co tangent, X times, Corsican two x And we&#x27;re supposed to use the fact that the limit SK goes is euros. I, Katie, but by Kinko&#x27;s first thing we&#x27;re gonna do is we&#x27;re gonna try to manipulate what we have inside the limit. So reversible since we&#x27;re looking for something like sine sine X divided by X. So we&#x27;re going to try to convert court engine into terms of sided co sign. So we know that the co tangent is just courtside X divided by acts. So we have this right here, and we also know that CO seek out to Rex is just one divided by side. And then now we&#x27;re gonna try to split our six x squared in such a way that whatever we have inside the sign is going to be found in the numerator. So, for example, here we have sine X, So we&#x27;re gonna pull in X from that 66 square Here we are here. We have to act. So we&#x27;re gonna pull it two x from the six x square. So now we&#x27;re left with three times. Co sign X multiplied by X divided by sine X multiplied by two x divided by sine two wings. So we&#x27;re gonna take our manipulation. But we did right here. I&#x27;m gonna plug it in to we&#x27;re gonna substitute it in toward limit as experts to zero. So now we have three times co sign next multiplied by another way to write X divided by sine X We can write it as the reciprocal of sine X, divided by x So here we have one divided by sine X divided by X And again we can also do that same trick in right It&#x27;s two x divided by sine toe Xs one provided by signed two extra biotic Plato And now we take the limit as excuse to zero. So the three years of constant so stays the same Now coastline X as X goes to zero. This becomes one because if we plug in zero, we get one fear That denominator becomes one. But we have one divided by one So that also this one again Here we have Wonderful. It will be one divided by one which is again one. So we have three times, one times one times one. So our final and source of the limiters exposed to zero this three.

Stephen Hobbs · Answer

plugging in Initially here we get 0/0. And so we&#x27;ll go ahead and re arrange and rewrite some terms. For instance, co sign Sorry. Co tangent is really the same thing as co sign of X over sign of X and CO seeking to X Well, we&#x27;ll keep green is the same thing as wand over sign of two X. Now we&#x27;ve been using rules involving X over sign of X and so on, and so that this is what we&#x27;re gonna do. Finally, we&#x27;re gonna go ahead and split up this expression here What will rewrite everything under me. So one way to do that, we could call it three times two x times x on the reason that I&#x27;ve chosen this is because if we looked at X over sign of X since experts zero, this would be one. Similarly, this two x divided by sine of two X would also give us one. And so ultimately, we&#x27;re really just left with. That&#x27;s three here. So this amendment evaluates 23

R M · Answer

number nine Limit existence Negative one of X Square Boss X That&#x27;s fine over its spread minus the X mind That&#x27;s who were which can be seen replied to extra plus five. But the blind by extra post one over its minus war on the lying X plus one so they can be eliminated and we can substitute with the negative one it iss ah functions that were negative one plus 54 and one minus worth is equal to negative negative five.

Foster Wisusik · Answer

Okay, This question wants us to evaluate the following limit. And to do this, our first step should always be the directly substitute and see if that works. So if we do that, we&#x27;ll plug in to and get four plus two minus six, divided by two minus two. And if we do this, we get zero divided by zero. And that means it&#x27;s an indeterminant form so we can do something to this limit to find the answer, and we have a quadratic on the top. So let&#x27;s factor that quadratic to see if that helps us. So we have X squared plus X minus six. So we need to numbers that multiplied a negative six and add toe one. So that&#x27;s positive. Three and negative, too. And if we look here now, we have a factor of X minus two on the top and the bottom. So they divide out, leaving us with the limit as X approaches to of X Plus three, which plugging in, we get three plus two, which is five, and that&#x27;s all right

R M · Answer

number 10 a limit off extents off X spirited minus war X Let&#x27;s war over X brand lost X Wyness seats, which compete Simplify toe X minus two times X minus two over X training When buying text. My misdoing, which can be eliminated and we can subsitute using X, is equal to two. So the fund and enter for that and would be, as you know, over five, which is.

Foster Wisusik · Answer

Okay, This question would like us to evaluate this limit. So, as always with these limit problems, we&#x27;re going to start just by directly plugging in. So to do this, plug in negative, too. And we get four plus two plus six for 12 on top, and then negative two plus two, which is zero on the bottom. So we know we have some kind of infinite limit or it doesn&#x27;t exist. So we need to test a point to sort of test the right and left hand limits to see if they match up. So we&#x27;re gonna test the limit as X approaches to her negative to from the positive side first. And we just care about the signs here. So if we have a number satellite Lee bigger than negative, too. We&#x27;re going to get a positive number on the bottom and then on the top, we&#x27;re also gonna have a positive number. So if we approach from the positive side, we have positive divided by positive, which is positive. Infinity. And then we did the same exact thing for the left hand side. So again the top remains unchanged because we have a positive plus a positive positive. Positive still and then But on the negative side or sorry. On the bottom side, we do get it negative because we get, like, negative 2.1 plus two, which is negative. So this sided limit is negative Infinity. So since the right side limit does not equal the left side limit, our limit does not exist.

Problem

Find the limits. $$\lim _{x \rightarrow 0} \frac…

Problem 30 Easy Difficulty

Find the limits.
$$\lim _{x \rightarrow 0} 6 x^{2}(\cot x)(\csc 2 x)$$

Answer

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Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals

Grace H.

Heather Z.

Kayleah T.

Caleb E.

Precalculus Review - Intro

Functions on the Real Line - Overview

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University Calculus: Early Transcendentals

Grace H.

Heather Z.

Kayleah T.

Caleb E.

Precalculus Review - Intro

Functions on the Real Line - Overview

Joel Hass, Christopher Heil, Przemyslaw BogackiUniversity Calculus: Early Transcendentals

Grace H.

Heather Z.

Kayleah T.

Caleb E.

Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals