- [Voiceover] So, let's see if we can figure out
what the limit as X approaches infinity
of cosine of X over X squared minus one is.
And like always, pause this video
and see if you can work it out on your own.
Well, there's a couple of ways to tackle this.
You could just reason through this and say,
"Well, look this numerator, right over here, cosine of X,
"that's just going to oscillate between
"negative one and one."
Cosine of X is going to be greater than or equal
to negative one, or negative at one is less than or equal
to cosine of X which is less than or equal to one.
So, this numerator just oscillates
between negative one and one as X changes,
as X increases in this case.
While the denominator here, we have an X squared,
so as we get larger and larger X values,
this is just going to become very, very, very large.
So, we're going to have something bounded
between negative one and one divided
by very, very infinitely large numbers.
And so, if you take a, you could say, bounded numerator
and you divide that infinitely large denominator,
well, that's going to approach zero.
So, that's one way you could think about it.
Another way is to make this same argument,
but to do it in a little bit more of a mathy way.
Because cosine is bounded in this way,
we can say that cosine of X
over X squared minus one
is less than or equal to.
Well, the most that this numerator can ever be is one,
so it's going to be less than or equal to one
over X squared minus one.
And it's going to be a greater than or equal to,
it's going to be greater than or equal to,
well, the least that this numerator can ever be
is going to be negative one.
So, negative one over X squared minus one.
And once again, I'm just saying,
look, cosine of X, at most, can be one
and at least is going to be negative one.
So, this is going to be true for all X.
And so, we can say that also the limit,
the limit as X approaches infinity of this
is going to be true for all X.
So, limit as X approaches infinity.
Limit as X approaches infinity.
Now, this here, you could just make the argument,
look the top is constant.
The bottom just becomes infinitely large
so that this is going to approach zero.
So, this is going to be zero is less than or equal
to the limit as X approaches infinity
of cosine X over X squared minus one
which is less than or equal to.
Well, this is also going to go to zero.
You have a constant numerator, an unbounded denominator.
This denominator's going to go to infinity,
and so, this is going to be zero as well.
So, if our limit is going to be between zero.
If zero is less than or equal to our limit,
is less than or equal to zero,
well then, this right over here
has to be equal to zero.