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## Analyzing limits numerically

# Approximating limits using tables

AP.CALC:

LIM‑1 (EU)

, LIM‑1.C (LO)

, LIM‑1.C.5 (EK)

## Video transcript

- 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.