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Numerade Educator

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Problem 54 Hard Difficulty

Let $ f(x) = [ \cos x ] $, $ -\pi \le x \le \pi $.
(a) Sketch the graph of $ f $.
(b) Evaluate each limit, if it exists.
(i) $ \displaystyle \lim_{x \to 0}f(x) $
(ii) $ \displaystyle \lim_{x \to (\pi/2)^-}f(x) $
(iii) $ \displaystyle \lim_{x \to (\pi/2)^+}f(x) $
(iv) $ \displaystyle \lim_{x \to \pi/2}f(x) $
(c) For what values of $ a $ does $ \displaystyle \lim_{x \to a}f(x) $ exist?

Answer

a. see the graph
b. $\begin{array}{llll}{\text { (i) } 0} & {\text { (ii) } 0} & {\text { (iii) }-1} & {\text { (iv) DNE }}\end{array}$
c. Exceptions: $a=\pm \pi / 2$

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Video Transcript

this problem. Number fifty four of the Stewart character This eighth edition section two point three Let effort banks equal the greatest interject function. Thousand cosign x from the Domaine negative pie to pie. So exit detaining a pie and by party's gets the graph of the function f So please applaud and vice to sketch the function. This is what we would see from negative pie towards half of pine. The value of this That function is negative. One from the value of negative power half to zero value zero at X equals zero. The value is one from zero to power to value zero and then repair over to two pi. You thought he was negative. One ironies is plot to help us answer some limit questions for being you know, they were each limit of it exists. What is the limit is expert easier of F. Well, let's take a look at the graph. As we approached zero from the left, we see that we approach the value of zero. As we're preserving the right, we see the reproach of value zero as well. So even though the value at zero of this greatest interest, your function cosign of X is equal to one. The limit exists and is equal to zero because that's what the behavior of the function is pointing towards. So this part one is equal to zero. Part two. What is the limit is exporters power two from the left of this function F however, to is right over here as we approach it Private Tiffin left. We see that the value of the function is equal to zero Part three What is limited c approach pie or two from the right of this function F as we trace the function from the right towards power too, we are approaching or we are along the Michaels negative one line and that is what the value the limit is as you approach Piper two from the left or from the right. So negative one Part four. What is the limit? Is experts power for two of the function f Well, we found the limiters approached from the left zero from the loud. We also see that limits approach. Property from the right is negative one because these two limits do not agree with each other and they're not equal to each other. We say the limit does not exist as you approach power too, for the function X martine for values of aid is Lim delimit as experts is eight or the function exist well, we already looked at part or point zero and with the point power to we saw that the limit existed at zero. In fact, the limit exists everywhere across this line, we see that the result of the limited, not existing power too. And as you can see that there's also another jump in the function here on negative power too. So these are two exes ex values that we show meant from where the limit exists. But we see that since the's ah, functions are continuous and ah, in this case there is a hole but limit exist, said zero. The limit exists throughout the interval negative part of pine, except for the two points where X is equal to pi over to and negative power too. So they will say all values between negative pion time. But ex cannot be equal to positive ornate eoe eye or two. And there's the final in