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Problem 18 Medium Difficulty

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

$ \displaystyle \lim_{x \to 0^-}f(x) = 2 $, $ \displaystyle \lim_{x \to 0^+}f(x) = 0 $,
$ \displaystyle \lim_{x \to 4^-}f(x) = 3 $, $ \displaystyle \lim_{x \to 4^+}f(x) = 0 $, $ f(0) = 2 $, $ f(4) = 1 $

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Video Transcript

All right, Y'all were doing graphs today. So we are supposed to sketch the graph of a function satisfying all six of the following criterion one, the limit as we approach zero from the left is to to when we approached zero from the right, we get 03 of the limit as we approach four from the left, we get three for the limit as we approach four from the right, we get zero and then five and six being f of zero equals two and f of four equals one. Now, as you can see, I've already laid out my graph nice to label the important point. So I've kind of label all the important Y coordinates and X coordinates. So the very first thing I want to take care of any time, I'm trying to graft something like this is any information that they give me about the original function in particular, F of zero equals two, and f of four equals one. So I better have points at zero comma two and four comma one. Now the limit information kind of tells me about what the function does around those points. Remember limits are giving me information about what the points around the point we're caring about think we're doing So for instance, as we approach zero from the left and are very first guy right here, ffx thinks that you're going to land on to. So literally what this translates to is that as you walk up to X equals zero from the left hand side, hence a little negative right here, F of X says, man, you look like you're approaching the y value of two so we can do kind of anything out here so long as we're approaching the number to. So maybe something like that. Now the limit as we approach zero from the right hand side, hence the plus right here is zero. What that means is that ffx predicts as you walk up to zero from the right hand side, that you will land on the Y value of zero. So f of X might be doing something like this. Like Hey, we're about to land on zero now we're gonna do an open dot here. The reason we do an open dot is because we're not technically ever touching zero comma zero, but we approach real close to it. Hence this whole limit that we get right here. Now when we move over to four, as we approach four from the left hand side yet again, that's why we have that negative. Our function thinks that will land on the y value of three. So pick up where we left off and we go away up here yet again, an open dot remember limit means you get infinitely close to but you never touch that point. So we get that open dot. But notice also that when we approach four from the right hand side, meaning from the more positive side, our function thinks that we're going to land on zero. So wait a second, we do something like this. Yeah. So yet again, important things in this problem. Make sure you first off graph these points, they have to be on your graph no matter what. That's why we draw a dot at zero comma two and four comma one From there. We used our limit information as we approach zero from the left, we approached the y value of two as we approach zero from the right, we approached the Y value of zero. And then for four, as we approach four from the left, we approached the Y value of three. And as we approach four from the right, we approached the y value of zero, open dots anywhere that you didn't happen. The land on a closed dot, that's how you go and translate all this information into a graph and yours might have looked very different. By the way, there's no reason that I have this like upward inflection right here. It could have been going downward. That would only be determined by more information given. But hey, that's artist's interpretation with what they gave me