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  • - [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.

- [Voiceover] So we have g of x being defined

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10-3 分析不連續性的函數(Analyzing functions for discontinuities (continuous example) | AP Calculus AB | Khan Academy)

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