Problem

(a) Use a CAS to show that if $k$ is a positive c…

Problem 64 Easy Difficulty

(a) Show that $\lim _{x \rightarrow \pi / 2}(\pi / 2-x) \tan x=1$
(b) Show that
$$\lim _{x \rightarrow \pi / 2}\left(\frac{1}{\pi / 2-x}-\tan x\right)=0$$
(c) It follows from part (b) that the approximation
$$\tan x \approx \frac{1}{\pi / 2-x}$$
should be good for values of $x$ near $\pi / 2 .$ Use a calculator to find $\tan x$ and $1 /(\pi / 2-x)$ for $x=1.57 ;$ compare the results.

Answer

a. $=1$
b. $=0$
c. $\tan (1.57) \approx \frac{1}{\frac{\pi}{2}-1.57}$

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Calculus 1 / AB

Calculus Early Transcendentals

Chapter 3

TOPICS IN DIFFERENTIATION

Section 6

L Hopital's Rule; Indeterminate Forms

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Video Transcript

talk about questioning the 64. We gotta show that limit extending towards by over two by my by over two minus extreme Stan axes one. So let's write it. US. Limit existent in towards power To, uh, we know that there is nothing but one walk caught, in fact, and it's nothing but sign over costs, so that can be it on a sine x over Cossacks. Yeah. Now limit exist. Ending towards by over two, uh, to spy over to minus extreme in Saturday's sine X remains as looters. No cause can better than us cause excess sign 90 minus X. We know that so we can write it like this. Another way to right. This is limit extending towards power toe sign extra means as it is. While this comes in the denominator off, sign off by over two minus x o x. So this becomes sign off by over two minus X over five or two minus X. Now, if you make a substitution off, let's say pi over to S t. Then if exists, tending towards by over two. Uh, in fact, let's make a substitution off. Be as by over two minus X. If you do that then since exist, tending towards spy over to then Definitely t must tend towards, uh, zero. Because when we put excess power to hear, he becomes zero. So the limit in terms off T is limited. E is tending towards zero. Here we have sign off excess replaced by 90 minus t. So here we have 90 ministry over a pie over to minus X is replaced Party. So we have scientists over. This comes out as t Now, if you place the limit we have, let's drag this number down if you place the limit Now we have signed five or two minus zero that assigned 90 which is one and sign 0/0 is also one. As for the former off the limits. So the answer for this one is one which is exactly what we had toe short are in part B. This will be used in parties will struggle over down in part B. We have to prove that limit exist. Ending towards by or two one over by over two minus sex minus panix is equal to zero. We have to show this. So let Z try to simplify this, uh Canberra denies limit Texas tending towards five or two This comes us by over two minus X here will be here to be one minus by over two 10 x plus X panics if you place access by over two. Ah, well, now it is better to make a substitution again, just like we did in party. So let's make a substitution off. Ah d as by over two minus x. So since existing indoors by over, two day tends towards zero. So the limit Let's let's try this little bit down. So if you make this substitution, then our limit terms out in terms off he becomes limit. He is tending towards zero one minus pi over toe tan off by over two minus. He is nothing but party. Plus, here we have access by over two minus t this will again big cart de on. This will just be being all right. So this comes out as limit. He is tending towards zero one minus five or toe be plus by over to court T minus t court t 40. So these two gets canceled. And what are we live? Further limit. He is tending towards 01 minus P caught t 40. So if you place the limit now, this can be infighter. Donna's limited. He's tending towards zero one minus t over 20 who are not anti over t o t over until a Matisse turning toward zero. Value is once one minutes 10010 So we can use, uh, h rules refuse the alleged rule than we have differentiation of 10 minus. Here we have 10 square T. Dante remains as it is different. Station off T is one minus 30. That means as a differentiation off minus six square T and differentiation off the denominator t is just one. All right, so let's try to simplify this further. So we have. Hmm. This is, uh, negative. Already outside. So negative goes inside. We have t sex square T minus tante over 10 square D. Now, this can be written as let's simplify the symptoms off sign and costs. So we have t costea over called Square T minus scientifically over. Cause be Let's shift this little bit down. Uh, hola. Science square T over. Cost 20. So we're going to simplify. The numerator will become t minus 70 times costea over. Call square. T over. This will be science square T over cause quality. So these two gets cancer And what are we left? Furthers are tee over Science square T minus. Things will be costea over Sign T and we have a limit off t uh, standing towards zero. Yeah, and fired Writing This won't really help us select straighter. And the simpler, simpler equations Simple, simpler, rational form art So steam minus scientific cost team over Science square D Placing t equals zero again gives us zero or zero selects again Use electoral So we have differentiation off t is one scientist Disasters, differentiation off causes minus sign plus causes as it is different. Station of sign is also caused over the transition of science where they will be to sigh Inti Costea So this comes orders one plus sine square t minus cost square t over toe scientific cause t So this comes out as, uh, scandal simplified us one minus called square T minus sine squared is cost duty and to scientific Costea signed to t This is again 0/0 for march. So we're gonna use electoral once again on definitely here we have limit t standing toward zero. So now if we differentiate this, we have zero minus minus two signed duty over to cause duty. So finally, we are getting liberties. Tending towards zero tanto tea and tan zero is nothing but zero. So the answer is zero, which is exactly what we needed to prove on Part C says that if the zero done tan X on pi over two minus x must be very same. So this is where the graph comes in, which we have toe prove we have to prove that 10 eggs. In fact, you have to give this a little bit like this. So we have to prove that Dan Eggs, Let's use a different color. Uh, we have to prove that Hear her tan x on by one over by over two minus sex at X equal to 1.57 Now, clearly 10 off 1.57 is coming as 1 to 2 55 point something. Well, one over by over two minus X is coming as 12 to 5.7658 so we can see that they are really very close. So this is a fairly good approximation off panics and won over by over two minus X

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