Analyzing limits numerically
Approximating limits using tables
- This video we're going to try to get a sense of what the limit as x approaches three of x to the third minus three x squared over five x minus 15 is. Now when I say get a sense, we're gonna 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 wanna try out is, well what happens to this expression when x is equal to three? Well then it's going to be three to the third power minus three times three squared, over five times three minus 15. So at x equals three, this expression's gonna be, let's see in the numerator you have 27 minus 27, zero. Over 15 minus 15, over zero. So this expression is actually not defined at x equals three. We get this indeterminate form, we get zero over zero. But let's see, even though the function, or 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 gonna set up a table. So let me set up a table here. And actually I wanna set up two tables. So this is x and this is x to the third minus three x squared over five x minus 15. And actually, I'm gonna do that again. And I'll tell you why in a second. So this is gonna be x and this is x to the third minus three x squared over five x minus 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 gonna, 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, you can go to two point nine and figure out what the expression equals when x is two point nine. But then we can try to get even a little bit closer than that, we could go to two point nine nine. And then we could go even closer than that. We could go to two point nine nine nine. And so 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. Now 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, those would be x values larger than three. So we could say three point one, but then we might wanna get a little bit closer, we could go three point zero one. But then we might wanna get even closer to three. Three point zero zero one. And every time we get closer and closer to three, we're gonna get a better approximation for, or we're gonna 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, two nine nine nine nine nine. Three point zero zero zero zero one. 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 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 one point eight. So is this equal to one point eight? As I said, in the future, we're gonna be able to find this out exactly. But if you're not sure about this you could try an even closer and closer and closer value.