(8) 2. Use the given graph of f to determine the value of...

(8) 2. Use the given graph of $f$ to determine the value of each quantity, if it exists. If it does not exist, explain why a) $\lim_{x \to -1^-} f(x)$ b) $\lim_{x \to -1^+} f(x)$ c) $\lim_{x \to 1} f(x)$ d) $\lim_{x \to 0} f(x)$ e) $\lim_{x \to \infty} f(x)$ f) $\lim_{x \to -\infty} f(x)$ g) $f(0)$ h) Equations of any VA or HA (if any exists)

Transcript

00:01 Okay, we're giving this graph and we're going to try and find all these things out about it.

00:06 Okay, the first one wants to know the limit is x approaches two from the left.

00:12 Okay, so that means we've got to get on the function, but we've got to be to the left of two, x equals two, because these are x values under here.

00:20 So here's x equals two right here, and then here's the left part of the function.

00:28 Okay, so we're on this function, and we're going toward two, what y value are.

00:33 Are we going to, what is the y value of that hole right there? well, the y value is 5, so that's the limit.

00:42 Okay, the next one says, now come at x equals 2 from the other side.

00:48 But remember, you have to be on the function.

00:50 So we have to be on this piece.

00:53 So we're going toward x equals 2 and what y value is that? well, that y value is 3.

01:01 So what is the limit as x approaches 2 of this function? well, we got five from one side and three from the other so the limit doesn't exist.

01:12 And then the last one says, what is the actual value of the function at 2, at x equals 2? what is the y value when x is 2? well, the y value is 4 because it's this point.

01:26 That's where the function is when x is 2 because it's colored in.

01:31 So the function value at 2 is equal to 4.

01:36 All right.

01:36 Now we can do the same thing, only we're coming at negative 2.

01:41 Okay, this one says, what's the limit as x approaches negative 2? and it doesn't say from the left or the right.

01:46 So we have to check both.

01:49 Okay, from the left, when x is going to negative 2, which is right here, we're on the function going toward that x value.

01:58 And we get y is 5.

02:01 And from the right, we also get y is 5.

02:05 So the limit is 5.

02:08 Then this says, what's the function value when x is negative 2? when x is negative 2, the place that's colored in is up here at 6.

02:18 So f of negative 2 is 6.

02:23 All right, what's the limit as x approaches negative 5? again, we've got to come from the left and the right.

02:29 So from the left, as we get close to x equals negative 5, we're going to y equals 2.

02:35 And from the right, also to y equals 2.

02:38 So the limit is two.

02:43 What's the function value at negative five? well, at negative five, there's a hole and there's nothing else.

02:49 So that function value doesn't exist.

02:52 That function doesn't exist at negative five because there's a hole.

02:56 Okay, so look back at negative two.

02:58 There was a hole there, but there was also a point where it was colored in, just not at the same place where the hole was.

03:05 Same as that positive two.

03:07 There's two holes and a colored in point.

03:10 So the function value is at the colored end point.

03:15 All right.

03:15 What's the limit as x approaches 4? positive 4.

03:21 Okay.

03:21 So from the left, we're on the function.

03:25 I'll just get another color going here.

03:27 On the function, i'm going toward x equals positive 4, and y is 1 there...

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