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Worked example: point where a function is continuous

In this video, we explore the limit of a piecewise function at the point where two cases of the function meet. By finding the left-hand and right-hand limits, we can determine if they're equal. If so, the limit exists at that point, and we've successfully analyzed the function's behavior.

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Video transcript

- [Voiceover] So we have g of x being defined as the log of 3x when zero is less than x is less than three and four minus x times the log of nine when x is greater than or equal to three. So based on this definition of g of x, we want to find the limit of g of x as x approaches three, and once again, this three is right at the interface between these two clauses or these two cases. We go to this first case when x is between zero and three, when it's greater than zero and less than three, and then at three, we hit this case. So in order to find the limit, we want to find the limit from the left hand side which will have us dealing with this situation 'cause if we're less than three we're in this clause, and we also want to find a limit from the right hand side which would put us in this clause right over here, and then if both of those limits exist and if they are the same, then that is going to be the limit of this, so let's do that. So let me first go from the left hand side. So the limit as x approaches three from values less than three, so we're gonna approach from the left of g of x, well, this is equivalent to saying this is the limit as x approaches three from the negative side. When x is less than three, which is what's happening here, we're approaching three from the left, we're in this clause right over here. So we're gonna be operating right over there. That is what g of x is when we are less than three. So log of 3x, and since this function right over here is defined and continuous over the interval we care about, it's defined continuous for all x's greater than zero, we can just substitute three in here to see what it would be approaching. So this would be equal to log of three times three, or logarithm of nine, and once again when people just write log here within writing the base, it's implied that it is 10 right over here. So this is log base 10. That's just a good thing to know that sometimes gets missed a little bit. All right, now let's think about the other case. Let's think about the situation where we are approaching three from the right hand side, from values greater than three. Well, we are now going to be in this scenario right over there, so this is going to be equal to the limit as x approaches three from the positive direction, from the right hand side of, well g of x is in this clause when we are greater than three, so four minus x times log of nine, and this looks like some type of a logarithm expression at first until you realize that log of nine is just a constant, log base 10 of nine is gonna be some number close to one. This expression would actually define a line. For x greater than or equal to three, g of x is just a line even though it looks a little bit complicated. And so this is actually defined for all real numbers, and it's also continuous for any x that you put into it. So to find this limit, to think about what is this expression approaching as we approach three from the positive direction, well we can just evaluate a three. So it's going to be four minus three times log of nine, well that's just one, so that's equal to log base 10 of nine. So the limit from the left equals the limit from the right. They're both log nine, so the answer here is log log of nine, and we are done.