WEBVTT

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- [Voiceover] So, let's
see if we can figure out

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what the limit as X approaches infinity

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of cosine of X over X
squared minus one is.

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And like always, pause this video

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and see if you can work
it out on your own.

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Well, there's a couple
of ways to tackle this.

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You could just reason
through this and say,

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"Well, look this numerator,
right over here, cosine of X,

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"that's just going to oscillate between

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"negative one and one."

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Cosine of X is going to
be greater than or equal

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to negative one, or negative
at one is less than or equal

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to cosine of X which is
less than or equal to one.

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So, this numerator just oscillates

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between negative one and one as X changes,

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as X increases in this case.

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While the denominator
here, we have an X squared,

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so as we get larger and larger X values,

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this is just going to become
very, very, very large.

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So, we're going to have something bounded

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between negative one and one divided

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by very, very infinitely large numbers.

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And so, if you take a, you
could say, bounded numerator

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and you divide that
infinitely large denominator,

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well, that's going to approach zero.

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So, that's one way you
could think about it.

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Another way is to make this same argument,

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but to do it in a little
bit more of a mathy way.

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Because cosine is bounded in this way,

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we can say that cosine of X

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over X squared minus one

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is less than or equal to.

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Well, the most that this
numerator can ever be is one,

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so it's going to be less
than or equal to one

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over X squared minus one.

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And it's going to be a
greater than or equal to,

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it's going to be greater than or equal to,

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well, the least that this
numerator can ever be

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is going to be negative one.

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So, negative one over X squared minus one.

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And once again, I'm just saying,

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look, cosine of X, at most, can be one

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and at least is going to be negative one.

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So, this is going to be true for all X.

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And so, we can say that also the limit,

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the limit as X approaches infinity of this

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is going to be true for all X.

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So, limit as X approaches infinity.

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Limit as X approaches infinity.

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Now, this here, you could
just make the argument,

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look the top is constant.

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The bottom just becomes infinitely large

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so that this is going to approach zero.

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So, this is going to be
zero is less than or equal

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to the limit as X approaches infinity

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of cosine X over X squared minus one

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which is less than or equal to.

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Well, this is also going to go to zero.

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You have a constant numerator,
an unbounded denominator.

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This denominator's
going to go to infinity,

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and so, this is going to be zero as well.

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So, if our limit is
going to be between zero.

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If zero is less than
or equal to our limit,

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is less than or equal to zero,

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well then, this right over here

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has to be equal to zero.