00:02
Okay, we want to talk about points of discontinuity.
00:05
So the first thing to do always is to factor anything that needs factored.
00:10
So the bottom here factors into x minus 3, x minus 2.
00:17
Okay, so you can see there's something going on at x equals 3 and x equals 2.
00:25
Okay, since the x minus 3s cancel out, that means there's a hole at that point.
00:32
So that leaves us 1 over x minus 2 okay so there's a hole at x equals 3 and let's see i'm not going to call f of 3 because f of 3 doesn't exist but we'll say the whole is at x equals 3 y equals well when you plug in 3 there 1 over 3 minus 2 you get 1 so whole at 3 1 okay, then also you have a discontinuity at x equals 2.
01:18
Okay, so that's the first part.
01:20
B.
01:21
And then at x equals 2, there's a vertical asymptote.
01:30
Okay.
01:31
And so since there's a vertical asymptote there, we have to look to see what's happening on each side.
01:37
So on the left, we'll take the limit as x approaches to.
01:47
From numbers smaller than 2.
01:54
So you get 1 over 0, but you're plugging in a number a little bit smaller than 2, like 1 .9.
02:01
So this is a little bitty negative number on the bottom.
02:05
So that's a positive divided by a negative, and so that is negative infinity.
02:12
Okay, so this notation right here means i'm getting a little tiny thing, but it's a little bit negative.
02:19
Okay, so we have 1 divided by a teeny tiny little negative number that gives us negative infinity.
02:26
Okay, then on the right, we're taking the limit as x approaches two from the right, 1 over x minus 2.
02:35
So you get 1 over 0, but this time we're plugging in numbers a little bit bigger than 2, like 2 .1.
02:41
So that's a little bitty positive thing, so that gives you positive infinity.
02:47
Okay, so here's what we got.
02:50
At 2, we have a vertical acetote.
02:54
As we approach from the left, it goes to next.
02:56
Negative infinity.
02:57
As we approach from the right, it goes to positive infinity.
03:02
And then also at 3 -1, we have a hole.
03:06
Okay, so here's the picture, basically, of what this one looks like.
03:13
Okay.
03:14
Okay, so i know it's a hole because we canceled the x -minus 3 out, and that leaves an asymptote at the other one.
03:21
All right.
03:22
So this one, x squared minus 3, x -minus 4 over x squared minus x -6.
03:31
Is that right? all right...