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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

A2 初級 美國腔

# 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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yukang920108 發佈於 2022 年 07 月 05 日