Name: 10-3 分析不連續性的函數（Analyzing functions for discontinuities (continuous example) | AP Calculus AB | Khan Academy）
Uploaded: 2022-07-05T13:38:34.000Z
Duration: 3 min 59 s
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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

when x is greater than or equal to three.

this three is right at the interface between

We go to this first case when x is between zero and three,

when it's greater than zero and less than three,

So in order to find the limit, we want to find

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 if they are the same, then that is going to be

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

When x is less than three, which is what's happening here,

So we're gonna be operating right over there.

That is what g of x is when we are less than three.

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,

So this would be equal to log of three times three,

when people just write log here within writing the base,

it's implied that it is 10 right over here.

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,

Well, we are now going to be in this scenario

right over there, so this is going to be equal

from the positive direction, from the right hand side

and this looks like some type of a logarithm expression

at first until you realize that log of nine

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.

as we approach three from the positive direction,

So the limit from the left equals the limit from the right.

They're both log nine, so the answer here is