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

This video we're going to try to get a sense of what the limit as x approaches 3 of ( x^3 - 3x^2 ) over ( 5x - 15 ) is. And 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 3. 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 ( 3^3 - 3 \times 3^2 ) over ( 5 \times 3 - 15 ). So at ( x = 3 ), this expression is going to be, and see, the numerator we 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 ( 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. And 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 3 from the left, from values that are less than 3. So for example, we 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 3, 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 3, 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 3; they're less than 3.

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 3. 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 3, ( 3.001 ). Every time we get closer and closer to 3, we're going to get a better approximation for—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. And you could keep going ( 2.99999999 ), ( 3.0001 ). 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 3 right over here from values less than 3. This is approaching the 3 from values larger than 3.

So now let's fill out this table, and I'm speeding up my work so that you don't have to sit through me typing everything into a calculator. So 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 and closer values.