Approximating limits | Limits and continuity | AP Calculus AB | Khan Academy

We're going to do in this video is see how we can approximate limits graphically and using tables. In the future, we're also going to be able to learn techniques where we're going to be able to directly figure out exactly what this limit is. But for now, let's think about how to approximate it, and it'll give—it'll build our intuition for even what a limit actually is.

So we want to know the limit as X approaches 2 of (\frac{x^3 - 2x^2}{3x - 6}). Now, the first thing you might want to check out is, well, what is this expression equal to when X is exactly equal to 2? We could do that by substituting x with 2. So it'll be (2^3 - 2 \cdot 2^2) over (3 \cdot 2 - 6). Well, this numerator over here is (8 - 2 \cdot 4 = 8 - 8), so that's going to be equal to zero. And then the denominator over here is (6 - 6), well that's zero, and we end up in the indeterminate form right over here. So this expression is not defined for (x = 2), but we can think about what does the expression approach as X approaches 2.

First, let's think about it visually. So if we were to graph it, and I graph this on the site Desmos, which has a nice graphing calculator, you see the curve (y) equals this expression right over here. So this is the curve of (Y = \frac{x^3 - 2x^2}{3x - 6}), and you can see at least everywhere where I've shown it here—and it's actually true—it's defined everywhere except for when X is equal to two. That's why we have this little gap over here showing that it is not defined.

What I want to do is approximate well, as X gets closer and closer to two, either from lower values of two or from larger values of two, what is the value of this expression or the value of this function approaching? At this level of zoom, it looks like it's approaching this value right over here. So as X gets closer and closer to two, it looks like our function is getting closer and closer to that value there, regardless of which direction we are approaching from.

So just approximating it visually right over here, let's see—this is zero, this is two, this is one right over here, this would be 1.5. At this level of zoom, it looks like it's about 1.3 or 1.4, so 1.3 or 1.4. Let's zoom in a little bit more, and if you have access to a graphing calculator or you go to a website like Desmos or Wolfram Alpha, you can zoom in further and further on this graph. So I encourage you to try that out yourself or do it with other graphs.

Let's zoom in even more. So once again, we're not defined at (x = 2), but here we get a slightly better read. Let's see, this is one, this is two—the value that we are approaching as X gets closer and closer to two, we're getting our values getting closer and closer to that right over there. If we look at what (y) value that is—let's see if this is split into 1, 2, 3, 4, 5—so this is 1.2 right over here, this is 1.4 right over here, so it looks like it's between. This would be 1.3 right over there, so it's a little bit more than 1.3. So approximately 1.3.

Now, I do that in a lighter color, so it's approximately 1.3 something it looks like. Let's zoom in even more to see if we can get an even better approximation. So now once again we're approaching that same value. We're not defined at (x = 2), but as we are approaching (x) equal to 2—let's see, let me get a darker color, so it would be right around there. This is—let's see, this is 1, 1.1, 1.2, 1.3, 1.4, 1.5—so once again, it looks like it's about 1.33 or 1.34, so I'd say maybe approximately 1.33.

So approximately 1.33 if I were to—looks like it might be approaching 1 and 1/3, but we don't know for sure. Remember, when you're trying to figure out a limit from a graph, the best you can really do here is just approximate, try to eyeball—well, the closer (x) gets to two, it looks like this function is approaching this value right over here.

Now, another technique, which tends to be a little bit more precise, is to try to approximate this limit numerically. So let's do that. Let me get rid of these graphs here. We already got a sense of what the graphs can do for us—they got us to about 1.33, but now let's try to do it numerically. So I’m going to set up a table here, and I encourage you to do the same.

On this column, I'll have my (X), and on this column, I'm going to say—well, what is the expression (\frac{x^3 - 2x^2}{3x - 6}) equal to? We know that when (X) is exactly equal to two, this thing right over here isn't defined, but let's see what happens as we approach two.

So let's see what happens when we're at 1.9 or 1.99. Well, I'll do 1.99; I'll do 1.999. And we could also see from the other direction; we could say, well, what happens when we approach from at 2.1? See if both these values seem to be approaching something. This is approaching it from lower values of (X), and then we could say this is approaching it from higher values of (X)—we could say 2.1 right over here.

Let me get a calculator out and evaluate these, and I encourage you to do the same. Get a calculator out and see if you can evaluate these things. Alright, so let's see if we can evaluate it when (x) equals 1.9. It's going to be (1.9^3 - 2 \cdot 1.9^2) and divided by (3 \cdot 1.9 - 6), and that's going to be equal to—looks like it's about 1.23. So I'll just write approximately 1.23.

Now let’s try it with a much closer value of (x). That was just 1.9. Now let’s go to 1.999. So once again, we're going to have (1.999^3 - 2 \cdot (1.999^2)) and then divided by (3 \cdot 1.999 - 6). This is interesting—approximately 1.332.

So numerically, it seems I am approaching that same value or close to that same value that I was approaching graphically. We can also do it from values of (X) greater than two. Let me get my calculator back, and I'll do some of it, and I encourage you to finish this up on your own. You could even try (1.999999) to see what it actually is approaching.

So for example, if I wanted to try 2.1, that would be (2.1^3 - 2 \cdot (2.1^2)) divided by (3 \cdot 2.1 - 6), and I get approximately 1.47. So now you can think I'm getting closer and closer to 2 from values larger than two.

Let’s see if we seem to be approaching the same value. Now let's get even closer to two, 2.01, and I'll do this one, and then I'll leave it up to you to see if you can get even more precise. If we want to be super precise, let's do three zeros here—(2.01^3 - 2 \cdot (2.01^2)) divided by (3 \cdot 2.01 - 6), and I get approximately 1.3333.

Now, it does indeed look like we are approaching 1.3333, or close to it. It looks like we are approaching 1 and 1/3. But once again, these are just approximations, both through the table or graphically. If you want to find the exact value of the limit, there are other techniques, and we're going to explore those techniques in future videos.