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Limits at infinity of quotients with trig

Video transcript

so let's see if we can figure out what the limit is X approaches infinity of cosine of X over x squared minus 1 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 be that's just going to oscillate between negative 1 and 1 cosine of X cosine of X is going to be greater than or equal to negative 1 or negative at 1 it's less than or equal to cosine of X which is less than or equal to 1 so this numerator just oscillates between negative 1 & 1 as X changes as x increases in this case well 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 1 and 1 divided by very very infinitely large numbers and so if you take a you could say a bounded numerator and you divide by an infinitely large denominator well that's going to approach 0 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 Matthew way because because cosine is bounded in this way we can say we can say that cosine of X over x squared minus 1 is less than or equal to well the most that this numerator can ever be is 1 so it's going to be less than or equal to 1 over x squared minus 1 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 1 so negative 1 over x squared minus 1 and once again I'm just saying look cosine of X at most can be 1 and at least is going to be negative 1 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 is constant the bottom just becomes infinitely large so 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 and an unbounded denominator this the denominator is going to go to infinity and so this is going to be zero as well so if our limit has two is going to be between zero it 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 it has to be equal to zero