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# Limits at infinity of rational functions

Sal analyzes the limits at infinity of three different rational functions. He finds there are three general cases of how the limits behave. Created by Sal Khan.

Video transcript

Let's do a few more examples of
finding the limit of functions as x approaches infinity
or negative infinity. So here I have this
crazy function. 9x to the seventh
minus 17x to the sixth, plus 15 square roots of x. All of that over 3x to
the seventh plus 1,000x to the fifth, minus
log base 2 of x. So what's going to happen
as x approaches infinity? And the key here, like we've
seen in other examples, is just to realize which
terms will dominate. So for example,
in the numerator, out of these three terms,
the 9x to the seventh is going to grow much faster
than any of these other terms. So this is the dominating
term in the numerator. And in the denominator,
3x to the seventh is going to grow much faster
than an x to the fifth term, and definitely much faster
than a log base 2 term. So at infinity, as we get
closer and closer to infinity, this function is going
to be roughly equal to 9x to the seventh over
3x to the seventh. And so we can say,
especially since, as we get larger and larger
as we get closer and closer to infinity, these
two things are going to get closer
and closer each other. We could say this
limit is going to be the same thing as this limit. Which is going to be
equal to the limit as x approaches infinity. Well, we can just cancel
out the x to the seventh. So it's going to
be 9/3, or just 3. Which is just going to be 3. So that is our limit, as
x approaches infinity, in all of this craziness. Now let's do the same with
this function over here. Once again, crazy function. We're going to
negative infinity. But the same principles apply. Which terms dominate as
the absolute value of x get larger and
larger and larger? As x gets larger in magnitude. Well, in the numerator, it's
the 3x to the third term. In the denominator it's
the 6x to the fourth term. So this is going to be the
same thing as the limit of 3x to the third over 6x to
the fourth, as x approaches negative infinity. And if we simplified
this, this is going to be equal to the
limit as x approaches negative infinity of 1 over 2x. And what's this going to be? Well, if the denominator,
even though it's becoming a larger and larger
and larger negative number, it becomes 1 over a very,
very large negative number. Which is going to get us
pretty darn close to 0. Just as 1 over x, as x
approaches negative infinity, gets us close to 0. So this right over here,
the horizontal asymptote in this case, is
y is equal to 0. And I encourage you to graph
it, or try it out with numbers to verify that for yourself. The key realization here
is to simplify the problem by just thinking
about which terms are going to dominate the rest. Now let's think about this one. What is the limit of
this crazy function as x approaches infinity? Well, once again, what
are the dominating terms? In the numerator, it's 4x to the
fourth, and in the denominator it's 250x to the third. These are the
highest degree terms. So this is going to be the
same thing as the limit, as x approaches infinity, of
4x to the fourth over 250x to the third. Which is going to be the same
thing as the limit of-- let's see, 4, well I
could just-- this is going to be the same thing
as-- well we could divide 200 and, well, I'll just
leave it like this. It's going to be the
limit of 4 over 250. x to the fourth divided by
x to the third is just x. Times x, as x
approaches infinity. Or we could even say this
is going to be 4/250 times the limit, as x
approaches infinity of x. Now what's this? What's the limit of x as
x approaches infinity? Well, it's just going
to keep growing forever. So this is just going to
be, this right over here is just going to be infinity. Infinity times some
number right over here is going to be infinity. So the limit as x approaches
infinity of all of this, it's actually unbounded. It's infinity. And a kind of obvious way
of seeing that, right, from the get go, is to
realize that the numerator has a fourth degree term. While the highest degree
term in the denominator is only a third degree term. So the numerator is
going to grow far faster than the denominator. So if the numerator
is growing far faster than the denominator,
you're going to approach infinity
in this case. If the numerator is growing
slower than the denominator, if the denominator is growing
far faster than the numerator, like this case, you
are then approaching 0. So hopefully you find
that a little bit useful.