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Math Help - I'm not understanding this limit problem...

  1. #1
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    I'm not understanding this limit problem...

    lim of (n)sin(1/n) as n goes to infinity. I thought the limit would be 0 since sin[1/(really big number)] = 0, but the book says the limit is 1. how do you get the limit as 1? Thanks.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by virtuoso735 View Post
    lim of (n)sin(1/n) as n goes to infinity. I thought the limit would be 0 since sin[1/(really big number)] = 0, but the book says the limit is 1. how do you get the limit as 1? Thanks.
    Let z=\frac{1}n\implies n=\frac{1}{z}. Then as n\to\infty, z\to0. Thus, \lim_{n\to\infty}n\sin\frac{1}{n}=\lim_{z\to0}\fra  c{\sin z}{z}=\dots
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  3. #3
    Like a stone-audioslave ADARSH's Avatar
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    Quote Originally Posted by virtuoso735 View Post
    lim of (n)sin(1/n) as n goes to infinity. I thought the limit would be 0 since sin[1/(really big number)] = 0, but the book says the limit is 1. how do you get the limit as 1? Thanks.
    Dont forget this
    Its not just

    sin[1/(really big number)]

    But its

    (really big number) x (sin[1/(really big number)])

    So can't just do it that way

    However put 1/n =t

    So when n-> infinity , t-> 0
    Hence it becomes

    Lt_{t->0} (sin[t]/t) = 1

    For the geometrical proof of this
    Read this
    --thanks to Chris for it

    EDIT: Chris won!!
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