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Math Help - Powers of limits

  1. #1
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    Powers of limits

    Given that
    e=lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n

    Wikipedia's proof that (positive integer x)
    e^x=lim_{n \to \infty} \left(1+\frac{x}{n} \right)^n

    contains this step:
    \left(lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \right)^x =<br />
lim_{n \to \infty} \left( \left(1+\frac{1}{n} \right)^n \right)^x

    Is that allowed? Why?
    Thanks

    Edit
    \left(lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \right)^x = lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \cdot lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \cdots lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n (?=) lim_{n \to \infty} \left(1+\frac{1}{n} \right)^{nx}
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  2. #2
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    Quote Originally Posted by MSUMathStdnt View Post
    Given that
    e=lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n

    Wikipedia's proof that (positive integer x)
    e^x=lim_{n \to \infty} \left(1+\frac{x}{n} \right)^n

    contains this step:
    \left(lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \right)^x =<br />
lim_{n \to \infty} \left( \left(1+\frac{1}{n} \right)^n \right)^x

    Is that allowed? Why?
    Thanks

    Edit
    \left(lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \right)^x = lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \cdot lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \cdots lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n (?=) lim_{n \to \infty} \left(1+\frac{1}{n} \right)^{nx}

    Because the exponential function is continuous, thus \displaystyle{\lim\limits_{n\to\infty}(a^n)^x=(\li  m\limits_{n\to\infty}a^n)^x\,,\,\,0<a\in\mathbb{R}

    tonio
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