Show that $\displaystyle lim_{x-->0}$ $\displaystyle ln (1+x)/x $ = 1 from the definition of derivative.
I did not follow you!
$\displaystyle \frac{ln(1+x)}{x}=\frac{ln(1+x)-0}{x-0}$
Do you have problem in this one?
Substracting zero is allowed everywhere.
and I replaced $\displaystyle 0$ by $\displaystyle f(0)$
since $\displaystyle f(0)=ln(1+0)=ln(1)=0$
Please, explain more.