Intro to Limits w/ Tables

This video will  introduce the concept of a limit in calculus by exploring secant and tangent lines, tables of values, and one-sided limits to understand function behavior near specific points. 

Summary

In this video, we introduce the foundational calculus concept of a limit, starting with how the slope of a secant line approaches the slope of a tangent line as two points draw closer together. Through visual demonstrations and tables of values in Desmos, we illustrate how limits allow us to evaluate a function's behavior near a point—even when the function itself is undefined or has a hole at that exact location. Finally, the lesson covers formal limit notation, reading limit statements, and using left- and right-hand one-sided limits to determine if a overall limit exists.

Key Takeaways

• A secant line crosses a curve at two points, while a tangent line touches it at just one.

• A limit describes the target value a function approaches as the input variable gets closer and closer to a specific point without needing to equal it.

• A limit can still exist at a given x-value even if the function itself is undefined or has a hole at that exact location.

• You can estimate a limit by making a table of inputs that get progressively closer to the target value from both sides.

• Limits can be analyzed specifically from the left side or the right side.

• An overall limit exists only when the function approaches the exact same output value from both the left and right directions.

Transcript

All right guys, Mr. Antonucci here and in this video we're going to talk about introducing the idea of a limit. And the first thing we want to do is talk about the slope of the tangent line to a graph. But to talk about the slope of the tangent line, and remember tangent means to touch, we have to look first at the secant line to a graph.

I think it'd be better if we looked at Desmos to check this out. So here we have the graph of a function in blue and we have the secant line which goes through these two purple points. Secant means to cut. So this line cuts through the graph of the function. To get the tangent line, that's where the line just touches the curve and doesn't cut through it. If we can take this second point and slide it closer and closer to this first point, you could see that it approaches the tangent line. But if we go ahead and zoom in some more, we could still see that there's some space in between the curve there and the line. And we can always get a little bit closer. The problem is that when that second point drops right on the first point, you're trying to get the slope between the same two points and there is no longer a slope at that point. So that presents a problem. Thus, we have to investigate what we call the limit.

So we're back here on this page and the slope of the secant line, remember, is just change in y over change in x. This is just in function notation. Now, since the limiting position of the secant line is the tangent line, then the limit of the secant line's slope should equal the slope of the tangent line. And this is what we would write in symbols here. And this lim as x approaches c is read "the limit as x approaches c."

So with that being said, let's talk a little bit more deeply about what the idea of a limit is. So here is the actual notation, and you would read this "the limit as x approaches c of f of x is equal to the number L." In other words, what this is saying is we can make the value of the function as close as we want to to L by choosing x sufficiently close to c, but not equal to c.

So here's some graphical illustrations of this situation. The most common one is where you have a function and the point that you're trying to get the limit at is actually part of the graph. So you could see as we approach the value c from either side, the y-value ends up approaching whatever this y-value would be right here. Now, in this case, we have a hole where the function is defined at the point above the hole. But still, the limit would be the same because as you approach that value from the left and right, you still would approach the same y-value because you're looking at values close to, but not equal to c. And here, even if the function is completely undefined at that point, the limit still exists because the y-values approach the same y-value from the left and right for all x values close to, but not equal to c.

So what we want to do next is just read and interpret each limit statement here. This you would read as "the limit as x approaches -1 of (x² - 1) / (x + 1) is equal to the number two." This one you would just read "the limit as x approaches 2 of 5x is equal to the number 10." Pretty straightforward. No tricks or anything like that there.

The next thing we want to do is talk about how to actually investigate a limit. And one of the ways that we can do this is by creating a table of values to help us better understand what limits are. So what we're going to do is investigate each of these limits: 2x + 5 and (x² - 1) / x.

The first one, we're going to look at what happens as x approaches 2. So we'll jump onto Desmos and make a quick table of values. So here I'm going to put in a table. If we hit the plus here and table, then we get a table of values. We'll talk about this function in a minute. The first thing that we had was 2x + 5. So I'm going to write this as 2x + 5. So now if I put in a table of values, we want to see what happens as x approaches 2. So we're going to put in values a little bit less than 2. So let's start at 1, and then 1.5... adjust this a little bit... don't want to go just to 2 yet... 1.9, and then 1.99, and 1.999. And we could see that the y-value gets closer and closer to 9 the closer you get to an x-value of 2, as you approach 2 from values less than 2.

The other thing we want to do is approach it from values greater than 2. So if we start at 3 and then get progressively closer, like 2.5, 2.1, 2.01, 2.001, and you could see that the y-values again are getting closer and closer to 9. Now, this function is defined at x = 2, so you can get the exact value, but remember the exact value is not what we're after. We're after the limit as x approaches 2. Now it is 9, so we have that there.

The other one that we had was... now in this case we're going to look at the limit as x approaches 1. Now you can see a problem here: by plugging in 1, it's not going to work because you would have 0/0. So again, if we're going to take the limit as x approaches 1, we're going to look at values less than 1 and approaching 1, and values greater than 1 and approaching 1. So less than 1, we could start at 0 and then get closer to 1: 0.5, 0.9, 0.99, 0.999. You could see that the y-values get close to 2. But we also have to anticipate or see what's going to happen from the right-hand side of 1. So we'll start at 2, get a little closer: 1.5, 1.1, 1.01, 1.001. And you could see that these points kind of almost look like they're going to make a line. But if you zoom in, you'll see that as the x-value gets closer to 1 from both sides, the y-values are getting closer to 2. Now, if I put exactly 1 in, the function's undefined, which is one of the reasons why investigating limits is so important: because we can see what a function looks like it's going to do as you get closer and closer to a value where it may be undefined.

One other piece that we want to quickly talk about are one-sided limits. Now, we've already kind of talked about these informally, but a one-sided limit is when you look at what the limit is as x approaches c just from the left or just from the right. To understand what happens as x approaches c from the left, we use a little negative as the exponent, and from the right, a little positive on the exponent.

So what we want to do here is investigate the limit as x approaches 0 of sin(x) / x. Make sure you're dividing the sin(x) by x and not just the x. You have your graph here, that's going to be helpful. And we want to investigate the limit as x approaches 0 from the left and right. So we can make a table of values. Change this to f(x₁). And then we're going to go -1, -0.5, -0.1, -0.01, and you could see that the y-values are getting closer to 1 from the left. So we would say the limit as x approaches 0 from the left of f(x) is 1.

Same thing we do, but from the right: 1, 0.5, 0.1, 0.01, 0.001, and we could see that the limit from the right is also 1. Since these two have the same value from the left and right, that suggests that the regular limit as x approaches 0 is 1.

All right guys, that's it for this video. So, hope that was helpful to you. Take care.