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For the function $ f $ whose graph is given, state the value of each quantity, if it exists. If it does not, explain why.

(a) $ \displaystyle \lim_{x\to 1}f(x) $

(b) $ \displaystyle \lim_{x\to 3^-}f(x) $

(c) $ \displaystyle \lim_{x\to 3^+}f(x) $

(d) $ \displaystyle \lim_{x\to 3}f(x) $

(e) $ f(3) $

(a) As $x$ approaches 1 , the values of $f(x)$ approach $2,$ so $\lim _{x \rightarrow 1} f(x)=2$

(b) As $x$ approaches 3 from the left, the values of $f(x)$ approach $1,$ so $\lim _{x \rightarrow 3^{-}} f(x)=1$

(c) As $x$ approaches 3 from the right, the values of $f(x)$ approach $4,$ so $\lim _{x \rightarrow 3^{+}} f(x)=4$

(d) $\lim _{x \rightarrow 3} f(x)$ does not exist since the left-hand limit does not equal the right-hand limit.

(e) When $x=3, y=3,$ so $f(3)=3$

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University of Michigan - Ann Arbor

University of Nottingham

Boston College

So in this problem we're given this function F on this graph here, this graph grass to find a number of limits and a value. So the first one were asked to find the limit As X approaches one ah F of X. So we can see, first of all as f as X approaches one, whether it's from the left from the right, then what we have is the value of the function as the function is continuous right here. And this value of the function is actually appear at two and I draw on this a little bit better. I would have drawn it like that. Okay, it would be obvious. They're all right, yeah, B is the limit as X approaches three from the left of our function. So, as we approached three from the left over here, I mean, we're coming up this curve, this goes to a value on our function of one. All right. They were asked to find, I put this up here, the limit, His ex approaches three from the right of our function. And so now we're coming up on this curve from the right, which goes to this value here. So this is four, and d says the limit as X approaches three of f of X. Well, the problem here is The limit as x approaches three means the limit from the left and the right. Both have to be the same and they are not. So this does not exist as for this to exist limit from the left, this limit and this limit must be equal and they are not so therefore that limit does not exist. And then E says F at three. Well, the function of three is where that dot is right there, at dot right there, which is appear at three. And so therefore, we have the answer now To all five parts of this problem.

Oklahoma State University