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Math Help - limit involving logs

  1. #1
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    limit involving logs

    Can someone please help me with this problem?

    Prove the limit relation

    lim_x-->0 log(x+1)/x = 1 by using the definition of the derivative

    Thanks in advance
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  2. #2
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    Quote Originally Posted by DudenOxford View Post
    Can someone please help me with this problem?

    Prove the limit relation

    lim_x-->0 log(x+1)/x = 1 by using the definition of the derivative

    Thanks in advance
    Not sure what "by using the definition of the derivative" means in this context, but L'Hopitals rule works here:

    <br />
\lim_{x \to 0} \log(x+1)/x = {\lim_{x \to 0} d/dx (\log(x+1))}\over {\lim_{x \to 0}d/dx (x)} = \lim_{x \to 0} 1/(1+x) = 1<br />

    RonL
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  3. #3
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    Quote Originally Posted by DudenOxford View Post
    Can someone please help me with this problem?

    Prove the limit relation

    lim_x-->0 log(x+1)/x = 1 by using the definition of the derivative

    Thanks in advance
    No need to use L'Hopital's rule here.

    Consider the function f(x) = \ln (1+x).
    This function is differenciable at zero, and furthermore,
    f'(0)=1.

    Hence, by definition of derivative,
    \lim_{x\to 0} \frac{\ln(1+x) - \ln (1+0)}{x-0} = \lim_{x\to 0} \frac{\ln (1+x)}{x} = 1
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