calculus - Find the limit of $lim_{xto0}{frac{ln(1+e^x)-ln2}{x}}$ without L'Hospital's rule

I have to find: $$\lim_{x\to0}{\frac{\ln(1+e^x)-\ln2}{x}}$$

and I want to calculate it without using L'Hospital's rule. With L'Hospital's I know that it gives $1/2$.
Any ideas?

Answer

Simply differentiate $f(x)=\ln(e^x +1)$ at the point of abscissa $x=0$ and you’ll get the answer. in fact this is the definition of the derivative of $f$!!