Approximating limit from numeric table

This video we're going to try to get a sense of what the limit as ( x ) approaches 3 of ( \frac{x^3 - 3x^2}{5x - 15} ) is. When I say get a sense, we're going to do that by seeing what values for this expression we get as ( x ) gets closer and closer to three.

Now one thing that you might want to try out is, well, what happens to this expression when ( x ) is equal to 3? Well, then it's going to be ( \frac{3^3 - 3 \cdot 3^2}{5 \cdot 3 - 15} ). So at ( x = 3 ), this expression is going to be, see, in the numerator you have ( 27 - 27 = 0 ), over ( 15 - 15 = 0 ). So this expression is actually not defined at ( x = 3 ). We get this indeterminate form; we get ( \frac{0}{0} ).

But let's see, even though the function, even though the expression is not defined, let's see if we can get a sense of what the limit might be. To do that, I'm going to set up a table. So let me set up a table here, and actually, I'm going to set up two tables. So this is ( x ) and this is ( \frac{x^3 - 3x^2}{5x - 15} ).

And actually, I'm going to do that again and I'll tell you why in a second. So this is going to be ( x ) and this is ( \frac{x^3 - 3x^2}{5x - 15} ). The reason why I set up two tables, I didn't have to do two tables; I could have done it all in one table, but hopefully this will make it a little bit more intuitive. What I'm trying to do is, on this left table, I'm going to, let's try out ( x ) values that get closer and closer to three from the left, from values that are less than three.

So for example, we could go to 2.9 and figure out what the expression equals when ( x ) is 2.9. But then we could try to get even a little bit closer than that. We could go to 2.99, and then we could go even closer than that; we could go to 2.999.

One way to think about it here is, as we try to figure out what this expression equals as we get closer and closer to three, we're trying to approximate the limit from the left. So, limit from the left. And why do I say the left? Well, if you think about this on a coordinate plane, these are the ( x ) values that are to the left of three. But we're getting closer and closer and closer; we're moving to the right, but these are the ( x ) values that are on the left side of three. They're less than three.

But we also, in order for the limit to exist, we have to be approaching the same thing from both sides, from both the left and the right. So we could also try to approximate the limit from the right. And so what values would those be? Well, those would be ( x ) values larger than three. So we could say 3.1, but then we might want to get a little bit closer. We could go 3.01, but then we might want to get even closer to three: 3.001.

Every time we get closer and closer to three, we're going to get a better approximation, or we're going to get a better sense of what we are actually approaching. So let's get a calculator out and do this. You could keep going 2.999999, 3.1.

Now, one key idea here to point out, before I even calculate what these are going to be, sometimes when people say the limit from both sides, or the limit from the left or the limit from the right, they imagine that the limit from the left is negative values and the limit from the right are positive values. But as you can see here, the limit from the left are to the left of the ( x ) value that you're trying to find the limit at. So these aren't negative values; these are just approaching the three right over here from values less than three.

This is approaching the three from values larger than three. So let's see, let's calculate the value of this expression when ( x ) is equal to 2.9.

So we have ( \frac{2.9^3 - 3 \cdot 2.9^2}{5 \cdot 2.9 - 15} ) is equal to — let's see — I got about 1.682. So now let's keep going.

If we do ( 2.99^3 - 3 \cdot 2.99^2 ) over ( 5 \cdot 2.99 - 15 ) is equal to approximately 1.788. And now let's get even closer to three. Remember, we're now approaching three from the left, so ( 2.999 ) is even that much closer to three.

So let's get the calculator out again. Now we get ( 2.999^3 - 3 \cdot (2.999)^2 ) divided by ( 5 \cdot 2.999 - 15 ) is equal to approximately 1.7988. So it seems to be getting closer and closer to 1.8.

You could even verify — I tried ( 2.9999 ) to see if it's getting closer to 1.8. So this is our limit from the left. Now we could try the limit from the right.

So what happens when we try 3.1? ( \frac{3.1^3 - 3 \cdot 3.1^2}{5 \cdot 3.1 - 15} ) is equal to 1.212.

All right, let's keep going. This is something strangely fun about this. All right. So let’s see when ( x ) is 3.01. So now we're approaching ( x = 3 ) from the right, from values that are larger, but we're getting closer and closer and closer.

We're approaching from the right, but we're moving towards the left, towards ( x = 3 ), without equaling ( x = 3 ). So now let's do ( 3.01^3 - 3 \cdot (3.01)^2 ) divided by ( 5 \cdot 3.01 - 15 ) is equal to approximately 1.812.

It does look like we're approaching 1.8 from above now, but let's just get even closer — ( 3.1 ). If you're unsure, you could just keep getting closer and closer and closer.

So ( 3.1^3 - 3 \cdot (3.1)^2 ) divided by ( 5 \cdot 3.01 - 15 ) is equal to roughly 1.81. So approximately 1.81.

Based on what we're seeing here, I would make the estimate that this looks like it's approaching 1.8. So is this equal to 1.8? As I said, in the future, we're going to be able to find this out exactly.

But if you're not sure about this, you could try even closer and closer values. Actually, just for kicks, let's just do a super close value. Let’s try it.

Let's do ( 2.99999 ). Let's see what that gets us. So we're getting really close; we're approaching from the left, and we're getting really close to three now. So let me get the calculator out here.

Okay, so how many is that? That is approximately 1.8. If we round it to the nearest thousandths, this is going to be 1.8. So, yeah, if we round it to the nearest thousandths, this is going to be 1.8.

So we're getting very close here to 1.8. If you're not feeling good about that, you could go even closer and closer and closer to three to feel better and better and better about it.

Once again, we're approximating, we're seeing what is the behavior as we get closer and closer to three from the left and from the right. Not only is this valuable for calculating limits — and sometimes you would have to calculate them numerically like this — but it also gives you an intuition for what a limit is all about.