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Math Help - Proof with limits

  1. #1
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    Proof with limits

    Prove that the lim as x approaches infinity of x^n/n^k=+infinity if x>1
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    do you mean x, not k?
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Snooks02 View Post
    Prove that the lim as x approaches infinity of x^n/n^k=+infinity if x>1
    Lets assume that you mean x and not k as the other poster said...then you want to show that

    \lim_{n\to\infty}\frac{x^n}{n^x}=\infty

    Consider that \frac{d^{x+1}}{dn^{x+1}}\bigg[x^n\bigg]=\ln(x)^{x+1}x^n

    and \frac{d^{x+1}}{dn^{x+1}}\bigg[n^x\bigg]=x!

    So now consdier that

    \lim_{n\to\infty}\frac{x^n}{n^x}=\frac{\infty}{\in  fty}

    Therefore L'hopital's may be applied...but we see that this also gives infinity over infinity...this continues until the x+1th derivative where we see that

    \lim_{n\to\infty}\frac{x^n}{n^x}\underbrace{=\cdot  s=}_{x \text{ times}}\lim_{n\to\infty}\frac{\ln(x)^{x+1}x^n}{x!}

    Which clearly equals infinity.
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