In the following exercises, use the graph of the function y=f(x) shown here to find the values,

In the following exercises, use the graph of the function...

Key Concepts

Limit of a Function

This concept involves determining the value that a function approaches as the input nears a specific point. It is fundamental in calculus as it forms the basis for defining continuity, derivatives, and integrals by examining the behavior of functions very close to a point rather than at the point itself.

One-Sided Limits

One-sided limits refer to the behavior of a function as the input approaches a specific point from one side (either from the left or the right). Analyzing both one-sided limits is crucial, since for the general limit to exist at a point, the left-hand limit and the right-hand limit must both exist and be equal.

Graphical Analysis of Limits

Graphical analysis involves using the visual representation of a function to estimate or determine the limit at a point. By examining the behavior of the function's graph near the point of interest, one can infer the value that the function is approaching, even if the function is not explicitly defined at that point.

Transcript

00:01 Okay, so we want to use the graph of the function y equals f of x to find the values.

00:06 Well, in this problem i can't exactly see the graph.

00:09 I don't think it copied over, but i'll give you a general idea of how to figure this out.

00:14 The limit as x approaches negative 2 of f of x.

00:20 How do we do that? so again, i can't see the function here that i think you were given.

00:26 I don't think it copied over properly, but i'm going to show you how to do this.

00:30 So we're going to do that with a sample function.

00:34 Sample function f of x.

00:38 So we'll have x values of negative 2.

00:42 And i'll run you through all of the possible options, how this function could turn out.

00:50 Ok, this is option one.

00:53 Ok, what is this limit? let's say this y value here is 4.

00:59 Well, to do that, we can take the limit from the left of x approaches negative 2, which is, if we're approaching from the left here as x approaches negative 2, we're approaching this y value of say, what is that 3? sure.

01:21 And then we'll take the right value.

01:24 The limit as x approaches negative 2 from the right is going to be also 3.

01:31 So what does this mean? this means if limits from both sides match as they do that the limit as x approaches negative 2 equals 3.

01:48 Okay, that's option 1.

01:50 The limit from the left, so if you just trace up the graph, and if we want to mark this negative 2 area here, which is the x value we care about, if we just trace this f -of -x function and see as x approaches negative 2 right here, what y value does this function approach? from both the left and from the right, this function approaches three.

02:15 So that is situation number one.

02:19 Okay.

02:20 Situation number two.

02:24 Because remember, i can't see your graph, so i have no clue what it actually looks like.

02:28 This is just me walking you through the possible options here.

02:34 Okay.

02:37 So say we have this function that goes like we.

02:41 And then it stops right here and picks up again, we like that.

02:47 Okay, how do we evaluate this function? we're going to use the same concept of approaching negative 2 from the left and approaching negative 2 from the right to figure out what the limit of negative 2 as a whole is.

03:05 So what do we get when we approach negative 2, x equals negative 2 from the left? and i forgot to draw these y values again, sorry.

03:16 We'll get that this is equal to negative 2 from the left, 3.

03:25 How about negative 2 from the right? coming in this way.

03:29 Looks like we get 2.

03:32 So when the limit from the left does not match the limit from the right, the limit does not exist.

03:43 The limit as x approaches negative 2.

03:47 Notice that there's no plus or minus after this because it's taking into account both sides.

03:52 It doesn't exist because the limit from the left doesn't match the limit from the right...

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