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Math Help - limit involving trig and L'H rule

  1. #1
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    limit involving trig and L'H rule

    Hi. Could somebody walk me through this one?:

    \lim_{x\rightarrow\infty}{x\,tan(\frac{1}{x})}

    I think that is equal to

    \lim_{x\rightarrow\infty}{\frac{x}{cot(\frac{1}{x}  )}}

    Which is of the form inf/inf. However, after two more attempted applications of L'Hs rule, I end up with a horrible mess that seems to be of the form 0*inf/inf, or something weird like that.
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  2. #2
    Member Goku's Avatar
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    Re: limit involving trig and L'H rule

    Let t = \frac{1}{x} so we get:

     \lim_{t \rightarrow 0} \frac{\tan(t)}{t} = \lim_{t \rightarrow 0}\frac{\sin(t)\cos(t)}{t} = \cos(0) = 1
    Last edited by Goku; February 28th 2013 at 09:57 PM.
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    Re: limit involving trig and L'H rule

    limit involving trig and L'H rule-temo-01-mar.png
    Thanks from infraRed
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    Re: limit involving trig and L'H rule

    It might help to look at the problem as  \frac{tan(\frac{1}{x})}{\frac{1}{x}}. Then if you differentiate using L'Hopital's Rule I think you should get  \lim_{x\rightarrow\infty} \frac{\frac{-1}{x^{2}} sec^{2}(\frac{1}{x})}{\frac{-1}{x^{2}}}
    Thanks from infraRed
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    Re: limit involving trig and L'H rule

    Quote Originally Posted by AZach View Post
    It might help to look at the problem as  \frac{tan(\frac{1}{x})}{\frac{1}{x}}. Then if you differentiate using L'Hopital's Rule I think you should get  \lim_{x\rightarrow\infty} \frac{\frac{-1}{x^{2}} sec^{2}(\frac{1}{x})}{\frac{-1}{x^{2}}}
    Yes, now I do seem to remember the professor giving an example similar to that. Thanks!
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