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Numerade Educator

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Problem 55 Hard Difficulty

If $ f(x) = [ x ] + [ -x ] $, show that $ \displaystyle \lim_{x \to 2}f(x) $ exists but is not equal to $f(2) $.

Answer

$0,$ so $\lim _{x \rightarrow 2} f(x) \neq f(2)$

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Video Transcript

this problem Number fifty five of this tour characterless eighth edition section two point three If. And that perfects is equal to the greatest interest your function of X plus the greatest interest in function of negative X show that the limit his expertise to of half exists but is not equal to the function after battle waiting too. So we will do this through example. So let's pick a value as he approached, too. We need to purchase two from the left. So let's say we approach two from the left and let's choose a value to represent this. Let's say we choose X is approximately one point nine nine. What is the value? The function at this point? Well, you have the interview or function at one point and then plus the other entered your function that negative point one point a name. Okay, the greatest interest or here would be one since as a reach to yet the greatest integer here would be in negative too. And so one when it's two causes negative one, let's see if we possibly get the same result from the rate as we approached you from the right we'Ll take the representative value of two point oh one. There's an example. What is the function evaluated at two point one? Well, that's the greatest. Entered your function of two Window one. Plus the greatest indigent enters your function of negative twenty one here. The greatest into Jerry's, of course, too, however, on the left side, on the negatives head. Once you cross that off the two point one, you are at the negative three region and this is negative or because of the limit from left equals limit from the right. We have confirmed it is, um, it does exist indeed. Now let's find out if. Why, Yeah, let's find out that it is not equal to effort, too. Oh, let's play enough to directly get there. Creative center. Jared, too Class the greatest entered your function to natives who, well, the greatest interest. Your function here is equal to two for the native to values, also negative, too. Two plus two gives us hero. Then we see that have integers and did your values. This function will be equal to zero and then everywhere else will be equal to NATO one. So we've confirmed that woman doesn't exist, but that is not equal to half of two