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Problem 17 Medium Difficulty

Sketch the graph of an example of a function $ f $ that satisfies all of the given conditions.

$ \displaystyle \lim_{x \to 3^+}f(x) = 4 $, $ \displaystyle \lim_{x \to 3^-}f(x) = 2 $,
$ \displaystyle \lim_{x \to -2}f(x) = 2 $, $ f(3) = 3 $, $ f(-2) = 1 $

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Video Transcript

satisfied five conditions F of three was 3. So when excess three to function value is three. So this point is on the graph of the function. Another condition f of negative two is one. So when X is negative 22 function value is one. So this point is on the graph of the function Next they gave us the limit of the function as X approached three from the positive side and then a separate limit As extra approached three from the left negative side. The limit of the function as X approached three from the positive side is four. So we need the graph of function approaching the value of four As X approaches three from the positive side but we need an open circle because the function does not actually hit for When X is three because we already know that FF three is 3. Now, the limit of the function as extra approach three from the left side from the negative side is too. So as X approaches three from the left, the function has to approach to once again it needs an open circle because it's not actually equal to two. When X is three, the function is equal to three when access three last but not least. Um We already mentioned that F of negative two is one. So this is the actual value of the function when X is negative two. Uh But another piece of information provided in the problem was the limit of F of X as X approached negative too. Now, since the limit of F of uh X as X approach negative two exists, that means it has to be the same limit whether we whether we approach negative two from the right side or we approach negative two on the left side. Um so the information said, the limit of F.A. extra approaches -2 is two. So as we approach negative two from the right side to function is approaching two. As we approach negative two from the left side to function needs to approach to. Okay, so we have the function approaching the value of two. Okay, from both sides of negative two from X is negative two. S extra approaches negative to the function is approaching to open circle because it doesn't actually equal to when X is negative 22 function is actually defined to be equal to one when X is negative two. So here is a graph of the function FA bex that satisfies all five conditions.