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Numerade Educator

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Problem 52 Hard Difficulty

Let
$ g(x) = \left\{
\begin{array}{ll}
x & \mbox{if $ x < 1 $}\\
3 & \mbox{if $ x = 1 $}\\
2 - x^2 & \mbox{if $ 1 < x \le 2 $}\\
x - 3 & \mbox{if $ x > 2 $}
\end{array} \right.$

(a) Evaluate each of the following, if it exists.

(i) $ \displaystyle \lim_{x \to 1^-}g(x) $
(ii) $ \displaystyle \lim_{x \to 1}g(x) $
(iii) $ g(1) $
(iv) $ \displaystyle \lim_{x \to 2^-}g(x) $
(v) $ \displaystyle \lim_{x \to 2^+}g(x) $
(vi) $ \displaystyle \lim_{x \to 2}g(x) $

(b) Sketch the graph of $ g $.

Answer

a. (i) 1 (ii) 1 (iii) 3 (iv) $-2$ (v)$-1$ (vi) DNE
b. see work for graph

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Video Transcript

This is problem or fifty two of the Stuart Calculus eighth edition Section two point three Let GM Back sequel This piece wise function made up of the first function. X if X is less than one in second function. Three. If X is equal to one, but their function to minus X squared, if one is less than X, is less than or equal to two in the last function, X minus three. If X is greater than two, evaluate each of the following If it exists part one of party The limit is X approaches one from the left of Jean. So as we approach one from the left, we are only concerned with the first function as we have not yet reached one, at which point we would switch to the second function Her for our solution to problem. Part one is that as we approach one from the left, we stay with this function and as we approach one, X equals one. The limit is equal to one part two limited expertise one no gene and then we have to look at both the left and the right. Lim left, left, left and we already solved in part one for the rec limit. We have to use this function here because this is where we are. Protein IX from the rate of one two x is greater than one. In this case, as we approach one two minus X squared approaches to minus one squared or one. So since the limit and six approaching, a warning from the rain is equal to warn her G is equal to one, and that would limit from the left is also equal to one. Then we say that the limit insects that produced one from G is equal to one. They both exist on their people to one what is G of one alone? Part three. We need to look at where X is equal to one. And there is this function on the value of the function at X equals one and says here is exactly equal to three. Her answer from birth three history part for the limiters. Expressions too. From the left of the function G, we look at which part of the function that coincides with here we're here. X is less than or equal to two. This will be the function as we approached two from the left. The function to minus X squared as we approach, too, should be two minus two squared or two minus four, which is equal to negative, too. Pipe part time. The limit is expertise to from the right of tea means that we have to look at the last function here since we're looking at, numbers are the largest them, too, as we approach, too, Disfunction expended three approaches a negative one, and so that is our limit for part five. Part six. Part six takes A look at the limit is he poached you from the left and the limited. He put Steven right Tina's although they both exist, but they have different values. We say that tournament, crying the limit of Part six, does not exist for the reason that limits from the left as you approach two does not equal delimit from the right. As he approached you finally, per p sketched the rest of the function. Here we have a plant showing each part X when X is less than one that is a straight line with the soul of one at X equals one. We ever Valley of three shown appear between one two. We have a new function to minus X squared, which is a downward facing problem. And then we see here that there's a jump provided that the limit is onyx. Aesthetics equals two where the function afterward is X minus three, and this is a graph of the function.