Answer
$\dfrac{\ln 3}{\ln 2}$
Work Step by Step
Consider: $\lim\limits_{x \to 0}f(x)=\lim\limits_{x \to 0}\dfrac{3^x-1}{2^x-1}$
WenNeed to check that the limit has an indeterminate form.
Thus, $f(0)=\dfrac{3^{0}-1}{2^{0}-1}=\dfrac{0}{0}$
The limit shows an indeterminate form. Thus, apply L-Hospital's rule: $\lim\limits_{a \to b}f(x)=\lim\limits_{a \to b}\dfrac{g'(x)}{h'(x)}$
Then
$\lim\limits_{x \to 0}\dfrac{3^x \ln 3}{2^x \ln 2}=\dfrac{3^{0} \ln 3}{2^{0} \ln 2}=\dfrac{\ln 3}{\ln 2}$