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## Analyzing limits at infinity

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# Limits at infinity of quotients with trig

## Video transcript

- [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.