evaluate-each-limit-if-it-exists-use-l-hospitals-rule-if-appropriate-lim-_x-rightarrow-pi-2-fracfrac

Question

Evaluate each limit (if it exists). Use $L$ Hospital's rule (if appropriate).
$$\lim _{x \rightarrow \pi / 2} \frac{\frac{\pi}{2}-x}{1+\sin x}$$

Carson Merrill · Accepted Answer

with this problem. We see that if we just plug in um pi over two directly, we would end up getting the indeterminant form. So we see that lope tells rule is applicable. So we&#x27;re going to take the derivative of the numerator. So the derivative of cosine X is going to be a negative Synnex and then the derivative of 1- Synnex is going to be negative cosine x. So based on this, we have negative Synnex over negative cosine X. That just equals tangent of X. Now that we have the tangent of X, we can evaluate this as X goes to pi over two From the right. So let&#x27;s find where x equals two pi over two. We see it right here. And based on this as it&#x27;s approaching from the right, it&#x27;s clear that we are approaching the value of um negative infinity because this is the graph of tangent of X. So, right, Because this is the tangent of X graph. We see that at pi over two, the tangent of poverty does not exist. So based on that, we see that our answer would be a negative infinity. As the final answer

Bryan Lynn · Answer

All right, so here we have the limit as X approaches pi over to from the positive side. Uh, co sign of X over one minus scientifics will start by evaluating the limit at pi over to. So if we do that and we presume that this is equal to the CO sign of pi Over to over one minus sign of pi over too. We end up with 0/1 minus one, which is 0/0 swill. Apply low Beatles rule here. Include that this is gonna be equal to the limit as X approaches pi over two of the derivative of co sign, which is gonna be negative Sign of X over the derivative of one minus sign of X, which will be so the one will drop out and we&#x27;ll get negative co sign of X. And, of course, then these two natives are going into simply cancel each other out and sign over. Co sign is, of course, just tangent. Right, So this is really equal to the limit as X approaches pi over two of the tangent of X OK, and so we can evaluate this and now Tangent of pi over to is undefined. Remember, we&#x27;re coming from the deposited end. And so if we wanted to remember what a graph of tangent looks like high over too. It&#x27;s gonna be something like that, right? So we are approaching negative infinity here. Therefore, the limit as X approaches however, to of tan X should be equal to negative infinity.

Jonathan Kerby-White · Answer

we have with limits as X approaches Pie co sign X minus one over X minus pi. The quantities were so to use our direct substitution just goods 0/0 so we can use lucky trolls Rule so women to Yugoslav images X approaches pie Negative sign looks over to times X minus pi. So if we put in pie just good 0/0. So he is a second application of Lotito&#x27;s room. So they get negative co sign works. Um, so this looks difficult, Teoh immediately evaluate So we can just distribute to the to to the X on a stoop. I so this looks much easier to differentiate Tu minus zero. So just to so we put in pie like Teoh co sign Pi over two, which equals 1/2.

Carson Merrill · Answer

for this problem. We&#x27;re going to be taking the limit as X goes to one of two, X minus one to the tangent kayaks over to. We&#x27;re gonna want to take the natural log on both sides. So to make things a little simpler and when we do that, we&#x27;re going to end up getting um the natural log of q minus X Divided by co tangent of high x over two. So now we can evaluate this at x equals one. Um but we&#x27;ll get 0/0 the indeterminate form. So now we&#x27;re going to use locales role. We&#x27;re going to take the derivative of the numerator and we get one over to mine attacks. And then we&#x27;re gonna take the derivative of asking me a negative actually. And we&#x27;re going to take the derivative the denominator and we end up getting a negative coast sequence squared. Okay. Hi X over two times. Hi over two as a result of this. When we evaluate it we end up getting chew over pie. But remember that that is the natural log of Y. Yeah. Equalling two over pie. So our final answer is going to be E to the two over. Hi, for the final answer.

Carson Merrill · Answer

The question is, find the following limit. Use the laptop through, or maybe a more elementary method. So, first, this is a one sided limit. And for one side limit, we can also use the lockdown through as long as this one satisfy the condition of loved US rule and the condition conditions of the laptops rule is that first, this function the top and about him about differentiable. And it is obvious because the both from the both functions elementary functions. So the top and the bottom a differentiable and second one is that the curative of the denominator is not zero. So first, let&#x27;s let&#x27;s find the directive of the of this one minus sine X, and we know that the directive of this is minus co sign X. And if you plug in pi over two here, that is minus pi minus coastline pi over two, which is zero. So in this way, this is sterile. So maybe in this way we cannot use the laptops rule. But, um, we can first write it as limit as X goes to hi or two, plus the curative of the top, which is minus Sigh X and divided by the curative of the bottom which is minus coz I x so minus minus has been canceled. So this is the limit X goes too high or too Plus sigh x divided by coz I X which is the limit as X goes to pie or two plus Condron hacks. So here the function becomes Tangent X And then in this in this way you can plug in pi over two and we know that the graph of, uh of tangent axis like this is like this Mhm, sure so And this side is pie or two and this function is periodic So the the next one will be like this. So here X goes to pi over two from the right that this X goes to Piau to like this. So as X goes to pi over two from the left from the right, the function goes to negative infinity. So we know that this limit is negative. Infinity

Problem

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

Problem 24 Easy Difficulty

Evaluate each limit (if it exists). Use $L$ Hospital's rule (if appropriate).
$$\lim _{x \rightarrow \pi / 2} \frac{\frac{\pi}{2}-x}{1+\sin x}$$

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Basic Technical Mathematics with Calculus

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

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