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Math Help - Limit of a function to the power infinite

  1. #1
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    Limit of a function to the power infinite

    What exactly is Limit of a function to the power infinite-lim.png?
    The possible answers are

    (a) e
    (b) e^2
    (c) e^3
    (d) \frac{1}{e}
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  2. #2
    MHF Contributor Amer's Avatar
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    Re: Limit of a function to the power infinite

    \lim _{x \rightarrow 0} e^{\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)^\frac{1}{x}

    since e is continuous
     e^{\lim _{x \rightarrow 0}\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)^\frac{1}{x}

    \lim _{x \rightarrow 0}\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)^\frac{1}{x}

    \lim _{x \rightarrow 0}\left(\frac{1}{x}\right)\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)

    \lim _{x \rightarrow 0}\left(\frac{\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)}{x}\right)
    0/0
    use lopital rule
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  3. #3
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    Re: Limit of a function to the power infinite

    Thanks. Some problems though...

    How exactly do you make e the base in the second step? (after "since e is continuous")?
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  4. #4
    MHF Contributor Siron's Avatar
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    Re: Limit of a function to the power infinite

    Quote Originally Posted by cosmonavt View Post
    Thanks. Some problems though...

    How exactly do you make e the base in the second step? (after "since e is continuous")?
    In general, u=e^{\ln(u)}, because y=\ln(u) \Leftrightarrow e^{y}=u \Leftrightarrow e^{\ln(u)}=u. So for this limit, let u=\tan\left(\frac{\pi}{4}+x\right)^{\frac{1}{x}}
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  5. #5
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    Re: Limit of a function to the power infinite

    Quote Originally Posted by Siron View Post
    In general, u=e^{\ln(u)}, because y=\ln(u) \Leftrightarrow e^{y}=u \Leftrightarrow e^{\ln(u)}=u. So for this limit, let u=\tan\left(\frac{\pi}{4}+x\right)^{\frac{1}{x}}
    Actually that was not my question. I know that u=e^{\ln(u)} but how did you make e the base in this:

     e^{\lim _{x \rightarrow 0}\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)^\frac{1}{x}
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  6. #6
    MHF Contributor Amer's Avatar
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    Re: Limit of a function to the power infinite

    since the exponential function is continuous
    lim f(g(x)) = f(lim g(x))
    if f is continuous at lim g(x)
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