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  1. #1
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    Limits

    Lim x tends to infinity ( e / ( 1 + 1 / x )^x ) ^x
    Pls help me...
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  2. #2
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    Re: Limits

    Quote Originally Posted by Abhi View Post
    Lim x tends to infinity ( e / ( 1 + 1 / x )^x ) ^x
    Pls help me...
    Is this \displaystyle \begin{align*} \lim_{x \to \infty}\left[\frac{e}{\left(1 + \frac{1}{x}\right)^x}\right]^x \end{align*} ?
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  3. #3
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    Re: Limits

    Quote Originally Posted by Prove It View Post
    Is this \displaystyle \begin{align*} \lim_{x \to \infty}\left[\frac{e}{\left(1 + \frac{1}{x}\right)^x}\right]^x \end{align*} ?
    yes
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  4. #4
    Member sbhatnagar's Avatar
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    Re: Limits

    Hello Friends!
    Solve \lim_{x \to \infty}[\frac{e}{(1+\frac{1}{x})^x}]^x

    \begin{align*} \lim_{x \to \infty}[\frac{e}{(1+\frac{1}{x})^x}]^x &= \exp \lim_{x \to \infty} x\ln[\frac{e}{(1+\frac{1}{x})^x}] \\ &=\exp \lim_{x \to \infty}x[\ln{e}-x\ln{(1+\frac{1}{x}})] \\ &=\exp\lim_{x \to \infty} [x-x^2\ln{(1+\frac{1}{x}})] \end{align*}

    For the limit \lim_{x \to \infty} [x-x^2\ln{(1+\frac{1}{x}})]
    \begin{align*} \lim_{x \to \infty} [x-x^2\ln{(1+\frac{1}{x}})] &= \lim_{x \to \infty} [x-x^2\left \{ \frac{1}{x}-\frac{1}{2x^2}+\frac{1}{3x^3}-... \right \}] \\ &=\lim_{x \to \infty}[x-x+\frac{1}{2}-\frac{1}{3x}+...] \\ &=\frac{1}{2} \end{align*}

    Therefore: \lim_{x \to \infty}[\frac{e}{(1+\frac{1}{x})^x}]^x=\exp\lim_{x \to \infty} [x-x^2\ln{(1+\frac{1}{x}})]=\exp(\frac{1}{2})=\sqrt{e}
    Last edited by sbhatnagar; November 29th 2011 at 05:04 AM.
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