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Derivative disguised as limit

  1. Sep 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose f is differentiable at a and f(a) =/= 0. Evaluate
    [tex]\lim_{n \rightarrow \infty}\left[\frac{f(a+\frac{1}{n})}{f(a)}\right]^n[/tex]

    2. Relevant equations
    None really

    3. The attempt at a solution
    Can someone check my argument:

    Since f is differentiable at a, f is continuous at a, so for sufficiently large n, f(a + 1/n) and f(a) have the same sign. Hence
    [tex]\lim_{n \rightarrow \infty}\left[\frac{f(a+\frac{1}{n})}{f(a)}\right]^n = \lim_{n \rightarrow \infty}\exp{\left(n\log{\left(\frac{|f(a+\frac{1}{n})|}{|f(a)|}\right)}\right)} = \exp{\left(\lim_{n \rightarrow \infty}\frac{\log{|f(a+\frac{1}{n})|} - \log{|f(a)|}}{\frac{1}{n}}\right)}.[/tex]
    This last limit in the argument of the exponential is the derivative of log|f(x)| at x = a, which is f'(a)/f(a). Thus, the original limit is simply e^(f'(a)/f(a)).
  2. jcsd
  3. Sep 2, 2009 #2


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    Homework Helper

    Looks perfect to me. :)
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