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Math Help - Proving that cosine diverges as x->infinity. Not sure on how to go about this

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    Proving that cosine diverges as x->infinity. Not sure on how to go about this

    I've to prove that as we take the limit of cosine as x goes to + infinity, the limit DNE. I've looked on different forums and sites, but all have methods I barely understand. I'm not very familiar with the Delta-Epsilon definition of limits. I've tried to look at the sticky at the top and try to understand but this is due tomorrow and I won't learn it at the moment unless it's the only way.

    What's a good way to start off a proof like this? I know the reasons why it doesn't exist, as it oscillates between -1 and 1, but I need to have a proof.

    Thanks
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    Re: Proving that cosine diverges as x->infinity. Not sure on how to go about this

    Quote Originally Posted by LightningZI View Post
    I've to prove that as we take the limit of cosine as x goes to + infinity, the limit DNE. I've looked on different forums and sites, but all have methods I barely understand. I'm not very familiar with the Delta-Epsilon definition of limits. I've tried to look at the sticky at the top and try to understand but this is due tomorrow and I won't learn it at the moment unless it's the only way.

    What's a good way to start off a proof like this? I know the reasons why it doesn't exist, as it oscillates between -1 and 1, but I need to have a proof.

    Thanks
    Are you familiar with the sequential characterization of limits?

    It states that if a limit exists it is independant of the sequence used to approch the point.

    Translation. For any two sequences

    x_n \to \infty \quad y_n \to \infty \quad f(x_n)=f(y_n)

    So if you can find two different seqences that go to infinity, but

    \cos(x_n) \ne \cos(y_n) that will show that the limit does not exist.
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