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Math Help - Prove that lim...=e

  1. #1
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    Prove that lim...=e

    Prove that

    \lim_{h \to 0}(1+h)^{1/h}=e

    Hints outlined for this question are as follows.

    Since \frac{d}{dx}\ln1=1,
    As h \to 0, \frac{\ln(1+h)-\ln1}{h}=\frac{\ln(1+h)}{h} \to 1
    A second hint is that if g is continuous at c and f is continuous at g(c), then f\circ g is continuous at c.

    You are not allowed to use L'Hospital's Rule to solve this question. This is one of the requirements, I'm afraid.
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  2. #2
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    Do you understand that \lim _{n \to \infty } \left( {1 + \frac{1}{n}} \right)^n  = e?
    If so let h=\frac{1}{n}.
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  3. #3
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    Quote Originally Posted by Plato View Post
    Do you understand that \lim _{n \to \infty } \left( {1 + \frac{1}{n}} \right)^n  = e?
    If so let h=\frac{1}{n}.
    Didn't quite get what you were getting at at first, but then I realized it. Thanks.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Runty View Post
    Prove that

    \lim_{h \to 0}(1+h)^{1/h}=e

    Hints outlined for this question are as follows.

    Since \frac{d}{dx}\ln1=1,
    As h \to 0, \frac{\ln(1+h)-\ln1}{h}=\frac{\ln(1+h)}{h} \to 1
    A second hint is that if g is continuous at c and f is continuous at g(c), then f\circ g is continuous at c.

    You are not allowed to use L'Hospital's Rule to solve this question. This is one of the requirements, I'm afraid.
    Let 1=\lim_{x\to0}\frac{\ln(1+x)}{x}=\lim_{x\to0}\ln\l  eft(\left(1+x\right)^{\frac{1}{x}}\right)=\ln\left  (\lim_{x\to0}\left(1+x\right)^{\frac{1}{x}}\right). The last part follows from the hint stuff. The conclusion follows by exponentiation.
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