Limits at infinity of quotients with trig
About this video
Sal finds the limit of cos(x)/(x^2 - 1) as x approaches infinity. First he reasons that a bounded numerator (between -1 and 1) divided by an ever-growing denominator goes to 0, then he makes this precise by trapping the expression between -1/(x^2 - 1) and 1/(x^2 - 1), which both go to 0, so the limit must be 0.
You'll be able to
- explain why a bounded numerator over an unbounded denominator approaches 0.
- use -1 <= cos(x) <= 1 to trap an expression between two simpler ones.
- conclude a limit is 0 when it is squeezed between two expressions that both go to 0.
Key words
- bounded
- Stuck between two fixed values; cos(x) is always between -1 and 1.
- oscillate
- To keep going up and down, like cos(x) swinging between -1 and 1.
- unbounded
- Growing without any limit, like x^2 as x gets larger.
- limit at infinity
- The value an expression approaches as x grows larger and larger without end.
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.
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