Answer
$\dfrac{\ln 3}{\ln 2}$
Work Step by Step
Let $\lim\limits_{x \to 0}f(x)=\lim\limits_{x \to 0}\dfrac{3^x-1}{2^x-1}$
But $f(0)=\dfrac{3^{0}-1}{2^{0}-1}=\dfrac{0}{0}$
This shows that the limit has an indeterminate form, so we need to apply L-Hospital's rule as follows: $\lim\limits_{m \to n}f(x)=\lim\limits_{m \to n}\dfrac{p'(x)}{q'(x)}$
This implies that
$\lim\limits_{x \to 0}\dfrac{(3^x) (\ln 3)}{(2^x) (\ln 2)}=\dfrac{3^{0} \ln 3}{2^{0} \ln 2}$
Thus, $\dfrac{(1) (\ln 3)}{(1) (\ln 2)}=\dfrac{\ln 3}{\ln 2}$
