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Problem 16 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) = 1 $, $ \displaystyle \lim_{x \to 3^-}f(x) = -2 $, $ \displaystyle \lim_{x \to 3^+}f(x) = 2 $, $ f(0) = -1 $, $ f(3) = 1 $

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Video Transcript

This is Problem number sixteen of Stuart eighth Edition, section two point two skits. The graph of an example of a function satisfies all of the given conditions. Okay, and we're going to read the conditions limit as experts. Zero of F is one. The limit is expert. History from the left calf is negative, too. The limit has exit purchased. Three from the right half is equal to two. The function effort evaluated at X equals zero is equal to make it one, and then the function of the function F evaluated in Mexico three is equal to one. So let's begin with one condition at the time, and we should be able to build on discuss any function that satisfies all of the conditions. The first limit, as we notice, is a limit that does not specify whether it's from the left or the right. And what this means is that we need to make sure we satisfy both delimit his expertise. They're from the left and the limit as X approaches zero from the right and also that these two limits must equal the same number. The only way that this limit exists limit is export zero because if the limit as export is here from the left exists, limit is expert zero from the right exists and that they're both equal. And in order for those two to be met in order for those conditions to be met and for this first given condition to be meant that limits should be equal to one. So what that means is that we need to choose a function that as the function approaches, X equals zero which is this y axis. It will approach one from the left and also one from the right. Something like this would be a good example. Any function dysfunction could have been decree are increasing upwards. Could have been a straight line. Any of those functions work as long as as you reach X equals zero on the functioning too. Approach one, but not necessarily beak equal to one. As we can see, we left this as a whole because that wasn't specified in this limit. And we see that for a dysfunction with this whole at one, we know that the limit for this function currently has experts. Zero is definitely one we'LL continue on with this function Maybe just making this part straight line for now, just to show that on dysfunction can have many different shapes. The next limit has expert just three. So we need this function just to approach that point. And this alone will satis fried this limit here because as we follow the function towards three from the left, we approached the value of negative too. And that has satisfied the second condition. The third condition is a requirement of the function to approach positive, too, when you approach X from direct. So just for simplicity will choose horizontal line, which is what the function would look like after X equals three and then show this whole at three. Comment tu at the point three two and this satisfies a third condition because the function behavior of the function as it approaches three from the right is that it approaches the value of two. The fourth condition has to do with the value of F when X is equal to zero. So if we look along this line, we see that currently there is no point to find when X is equal to zero. But this fourth condition gives us that defined point, it says that I was at X equals zero. Why's iCal too negative? One. So we will be filling that point in zero negative one, and we have met the fourth condition. Finally, the fifth condition specifies that the defined point at X equals three is equal to one. So look for their point. X equals three. The value is one we felt this point in because it is the defined point and we noticed that everything is consistent with all the given conditions. The limit is expert. Zero is equal to one clearly from the left and the right. The limit is expert degree from the left. His negative too. The limit is expert history from the right. It's positive too. And then we have the two to find points zero negative one and three positive one. And so this is an example or function that satisfies all of these given condition.