solve-the-given-problems-text-find-leftlim-_theta-rightarrow-pi-2sin-thetatan-theta-quad-text-see-ex

Question

Solve the given problems.
$$	ext { Find }\left.\lim _{	heta ightarrow \pi / 2}(\sin 	heta)^{	an 	heta} \quad 	ext { (See Exercise } 39 .ight)$$

Brian Sipko · Accepted Answer

we are asked to find the limit as X approaches. Prior to assign data raised the tangent data power. Now, if you plug in pi over to you right away, we find that we&#x27;re left with the form one to the infinity power. And this is one of our indeterminate forms so we can apply low guitars rule once we rearrange this. So first step it&#x27;s going to be letting why equal to the limit as X approaches pi over two of science data times tangent theater. Now we want to get rid of this tangent data as a power so we can take the natural log of both sides. So we can say that the natural log of why is equal to limit as X approaches pi over two of the natural log of science theater raised to the tangent fatal power. Now by our law algorithm rules, we can move this power the front so we can rewrite this as tangent theta times the natural log of science data. Now we want to rewrite this as an indeterminant product and to do that we can rewrite this as the natural log of sign theater divided by one over D. Tangent of theater. Now one over the tangent of data. This is the same as co tangent data. And now we have the form 0/0. So we can apply a little beetles rules the derivative of the numerator. We&#x27;re left with co sign of theta divided by sign theater and our denominator. We&#x27;re left with negative Kasey Kahne squared Ceta from here. We can rewrite this as co sign theater divided by sign theater divided by negative one over sine squared theta. Now what we can do here is we can pull this up and multiplied across. So we&#x27;re left with co sign theater divided by science either divided by one over and square data. So that&#x27;s just the same as times Science square theater. And we can&#x27;t forget are negative. Now we&#x27;re gonna cancel this science data in our denominator over there. That&#x27;s gonna cancel power here. This is going to leave us with negative cosign theta sign fever. We&#x27;re gonna copy this all the way down here. So we&#x27;re left with negative cosign theta sign set up. And if we plug in pi over two we&#x27;re left with zero. So we just solved for is the natural log of Y. So the natural love of why the secret is zero. And if you take the exponential of both sides we find that why is equal to E. 20 power Which is this one We know that. Why represents the limit Is experts five or two. So one is our final answer.

Zachary Mitchell · Answer

we have the limit as the data goes to zero of co sign Pi Seita divided by signed data. Now, because cosine is continuous everywhere, we can take the limit and move it into the co sign. And we can actually pull out the fit, pull out the pie, and then right this as a reciprocal. So let&#x27;s put the pie. And then we&#x27;re going to write this as one over the limit data goes to zero of sine theta divided by theta. Okay, and now we know that the limit us the limit of sign data divided by data is data goes to zero. Is one. This is something that we calculated earlier. So we get co sign of pi times one which is equal Thio negative one.

Michael Twiton · Answer

Let&#x27;s begin. Lee Rewriting dysfunction is a fraction. It&#x27;s 1/3 times sign of state, a divided way to co sign of things, and it&#x27;s also remind ourselves what these two trig functions look like. So here&#x27;s Stada he was play over to that&#x27;s trying to co sign. Coastline starts at a maximum of one, and it&#x27;s going down from there. So what happens is state approaches by over two from the positive direction this 1/3 stays put. Sign in black gives one at the top co sign gives a small negative number. Now division off the positive number, but a very small negative number means that the limits his negative infinity.

R M · Answer

broken Number 35. The limits of data tends to see you off status. Where? Over one minus school. Fine data. Which Ah, first we went a month. Apply bullfights by one plus course. Sign faith. Answer with a friend and answer for that would be data squared well, but one minus was signed. It&#x27;s where it&#x27;s time. One plus well, Science. Faith there Well, but later, over one one, Minus school Science Data, which is signed. This where faith there and the limit for status squared over sine squared later is equal toe one and the limit for one plus co sign data is one, and we treat is equal to zero cosigned. Zero resigns here is equal to one is one plus one since one plus one. Who&#x27;s the one time student?

Amy Jiang · Answer

forget the rest on the limits, as they approach is, they&#x27;re all of sign. There are over. Sign to data. Okay, we can rewrite this as actually, it&#x27;s multiplied by two later. And divide by data and divide by that as well. So we get the limits instead of approaches, uh, sighing data we&#x27;re gonna divide by data and also multiplied by data. Let&#x27;s do that. And then we&#x27;re gonna divide by two data and also multiply right to data all over sign, too. Okay. And then now plugging in zero arching limit as they&#x27;re ever. So we know that this is eagle one. This is also equal to one, and then we just gets used to terrify, and we&#x27;re left with one hand.

Issa Dababneh · Answer

So in this question, where as to determine the limited data goes to zero off sign Dana divided by sine two things. And we have to use the intimate relationship that has that limit. Its K goes to zero scientific one, and then directly. What it buys but we can get for using laws off limits is that the Limited&#x27;s cake goes to zero k, divided by sine K is equal to one, so their reciprocal is also called toe the limit of the reciprocal. So the first thing we&#x27;re gonna do is we&#x27;re gonna try toe ultra bracelet, manipulate what we have indolent. And yes, So the first thing we&#x27;re gonna do is we&#x27;re gonna split the limited to two parts. So what&#x27;s that? Sorry, but something we&#x27;re gonna spoke with on the inside of the limiting to two parts. So the first part is gonna be sign theater divided by one of the second part is signed wanted, but by scientific data, Well, we can do is now We have a beta of the inside of the side and we want a data that did not deter. So what we do is we multiply by, say it open data. So remember this is valid because fairly divided by fate has one same thing. We have a tooth ada in this sign right here in the development later. So what we can do is become multiply by two theater divided by two. So finally so now what we get We moved that date all the inside So we have side to side of the divided by data We have a food and on the outside, you hear we have a two divided by sine to fade and we have Ah, what divided by two theater on the outside And the first thing we notice is the weekend cat So the cedars So all this manipulation bodily Let this to say right, But we&#x27;re not done because the question tells us to determine that living So what we&#x27;re gonna put on the inside right now Inside the limit is one divided by two times scientifically divided by data times to theater divided by sine too. We&#x27;re gonna take the limit estate or goes to zero. So but 1/2 is a constant we can&#x27;t really taking the limit of it will only result in 1/2 Science fate over sailors once the living a better one two things are divided by sine to date of the second this for so we have 1/2 times one times one. This is just 1/2.

Problem

Solve the given problems. $$\text { Is } \lim _{x…

Problem 42 Easy Difficulty

Solve the given problems.
$$\text { Find }\left.\lim _{\theta \rightarrow \pi / 2}(\sin \theta)^{\tan \theta} \quad \text { (See Exercise } 39 .\right)$$

Answer

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Catherine R.

Anna Marie V.

Kayleah T.

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Basic Technical Mathematics with Calculus

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

Grace H.

Catherine R.

Anna Marie V.

Kayleah T.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus