00:03
All right.
00:04
So we have function absolute value of x squared minus nine.
00:08
And we gotta find where is this function? defensible and find the derivative and then part b graft.
00:15
Both those.
00:16
So we're going to start with what the absolute value function actually represents.
00:23
It's a combination of functions.
00:26
So you have one expenses greater than three.
00:31
You know, x squared minus nine.
00:36
When exit between three and negative three, you're gonna have x squared minus nine.
00:46
Negative.
00:48
And when x is less than negative three ganyu off x squared minus nine.
00:55
All right, then, using the definition of the limit, we have the our definition of the derivative.
01:07
You have purchase a f of x minus f of a over x minus.
01:19
Hey, now, just looking at the the f of x function that we have on the left, you can tell that that function changes at two points, it changes at three.
01:30
And edges and negative three, so we can anticipate derivatives.
01:36
Uh, not being, ah not having values at those points because of the function.
01:42
Changes on the derivative is gonna change.
01:47
So we'll test the derivative at x equals three first.
01:58
So the limit x approaches three like after 30 and then we plug in our function to get all right.
02:33
So in order for this limit to exist, the limit has to exist coming from the left and the right.
02:41
So we'll continue that over here with from the left when it's coming from the left.
02:52
We're gonna use this middle function because left means it's less than three.
03:07
Factoring dividing gives you expose three negative expects three and then plugging in three.
03:20
Getting out negative.
03:21
Six.
03:22
No, they're the same thing coming from the right.
03:30
Coming from the writing music x is gonna be greater than three.
03:32
So we use the top function x squared minus nine.
03:40
It's gonna come down to expose three plugging in three years uni at six.
03:45
So as you can see, these two are not the same.
03:52
So therefore the limit.
03:57
His ex approaches three x squared minus nine.
04:05
It does not exist, and therefore, if this doesn't exist, it means it's not differential at three.
04:14
And that was for equals.
04:16
Three.
04:17
Now we'll examine a equals negative three approaching from the left.
04:36
That means it's less a negative three system with that first function or 1/3 function right there.
04:46
Watch the sign right here because you're doing x minus a.
04:49
So it's gonna be positive.
04:50
Three plugging in negative three.
04:57
Gonna get negative.
04:59
Six.
05:07
Right.
05:07
So from the right means it's greater than *** three.
05:09
It's gonna be the middle function right there.
05:22
It's a negative.
05:24
X minus three.
05:26
Plug in bigger three.
05:28
You're going to get positive...