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Limits - composed functions

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  1. Jul 15, 2016 #1

    Rectifier

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    The problem
    $$ \lim_{x \rightarrow \infty} \frac{(\ln x)^{300}}{x} $$

    The attempt
    ## \lim_{x \rightarrow \infty} (\ln x)^{300} = \infty## since ## \lim_{x \rightarrow \infty} f(x) = A## and ## \lim_{x \rightarrow \infty} g(x) = \infty ## thus ## \lim_{x \rightarrow \infty}f(g(x)) = A ##.

    ## f(x) = x^{300} ##
    ## g(x) = \ln x ##

    ## \lim_{x \rightarrow \infty} \frac{1}{x} = 0 ##

    So in the end I get ## " 0 \cdot \infty " ##. Which is not an acceptable solution.
     
  2. jcsd
  3. Jul 15, 2016 #2

    Math_QED

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    Do you know l'Hospital's rule?

     
  4. Jul 15, 2016 #3

    Rectifier

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    No, I don't. We are supposed to solve it without it at this point.
     
  5. Jul 16, 2016 #4

    micromass

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    Write ##x=e^y##.
     
  6. Jul 16, 2016 #5

    Rectifier

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    I solved it by setting ##x=t^{300}## but your approach is even better.
     
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