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Math Help - How do you justify this limit?

  1. #1
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    How do you justify this limit?

    let {xn} as n tends to infinity be defined by xn=cos(n)/n. The limit is 0, but is the only way to justify this to use the definition of a limit?
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  2. #2
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     -1 \leq cos(n) \leq 1

     \frac{-1}{|n|} \leq \frac{cos(n)}{n} \leq \frac{1}{|n|}

    as n tends to infinity, you can take n > 0

     lim_{n\to\infty}\frac{-1}{n} = 0 and lim_{n\to\infty}\frac{1}{n} = 0

    By the squeeze thereom \lim_{n\to\infty}\frac{cos(n)}{n} = 0
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by amm345 View Post
    let {xn} as n tends to infinity be defined by xn=cos(n)/n. The limit is 0, but is the only way to justify this to use the definition of a limit?
    Usually the definition of the limit is the ultimate authority on a limit's value. How about using the squeeze theorem.

    \frac{-1}{n}\le x_n\le\frac{1}{n} for all permissable values. What can you say now?
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  4. #4
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    Thank you both! Would I then be able to something similar to show that the limit of (1+1/n)^(n+1) is e?
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by amm345 View Post
    Thank you both! Would I then be able to something similar to show that the limit of (1+1/n)^(n+1) is e?
    Well, that is a bit of a more tricky situation. You can bound it above and below, but this isn't exactly helpful. Do you know that \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=e. If so, just split your limit into \left(1+\frac{1}{n}\right)^n\left(1+\frac{1}{n}\ri  ght)....and then.............what?
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  6. #6
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    ahh I see. perfect! thank you again!
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