# limit involving logs

• May 23rd 2007, 02:55 AM
DudenOxford
limit involving logs

Prove the limit relation

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

• May 23rd 2007, 04:03 AM
CaptainBlack
Quote:

Originally Posted by DudenOxford

Prove the limit relation

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

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

$\displaystyle \lim_{x \to 0} \log(x+1)/x =$ $\displaystyle {\lim_{x \to 0} d/dx (\log(x+1))}\over {\lim_{x \to 0}d/dx (x)}$$\displaystyle = \lim_{x \to 0} 1/(1+x) = 1$

RonL
• May 24th 2007, 05:31 PM
ThePerfectHacker
Quote:

Originally Posted by DudenOxford

Prove the limit relation

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

Consider the function $\displaystyle f(x) = \ln (1+x)$.
$\displaystyle f'(0)=1$.
$\displaystyle \lim_{x\to 0} \frac{\ln(1+x) - \ln (1+0)}{x-0} = \lim_{x\to 0} \frac{\ln (1+x)}{x} = 1$