00:01
So in this question, we want to find each x value at which the function f is discontinuous.
00:06
And for each x value, we're going to determine whether f is continuous from the right, from the left, or neither.
00:13
So let's take a look at this function f of x.
00:17
It's piecewise defined.
00:18
It's x plus 8 when x is less than 0.
00:22
It's e to the x when x is between 0 and 1.
00:26
And it's 8 minus x if x is greater than 1.
00:30
So let's check about continuity at the first break point at x equals 0.
00:38
So let's see.
00:40
Let's look at the limit as x approaches 0 from the left of my function, and the limit as x approaches 0 from the right of my function.
00:56
Well, if x is approaching 0 from the left, which piece of function am i using? i'd be using the piece where x is less than zero.
01:10
I'd be using x plus eight.
01:14
To evaluate that limit, what do i do? i plug in, i get zero plus eight.
01:22
That's eight.
01:24
Now, let's look at the limit as x approaches zero from the right of my function.
01:31
If x is approaching zero from the right, which piece of function am i using? i'm using the piece where x is just a little bit bigger than zero.
01:41
I'm using e to the x.
01:44
To evaluate that limit, i plug in, i get e to the zero power, which is one.
01:52
Note that these two numbers are not equal to each other, and so we have a discontinuity at x equals zero.
02:02
Now, since the equality was attached to the the right hand side of 0, we would say this was continuous from the right.
02:19
Now, let's look at what happens at x equals 1.
02:25
So we're going to look at the limit as x approaches 1 from the left of f of x and the limit as x approaches 1 from the right of f of x.
02:40
Now if x approaches 1 from the piece of function am i using? i'm using the piece where x is smaller than one.
02:49
In this case, that's e to the x.
02:53
I plug in, i get e to the first power, which is e.
02:59
Then i look at the limit as x approaches one from the right of f of x.
03:05
If x is approaching one from the right, which piece of function am i using? i'm using the piece where x is bigger than one.
03:13
I'm using 8 minus x...