Answer
$(\dfrac{\ln 3}{\ln 2})$
Work Step by Step
L'Hospital's rule states that $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{A'(x)}{B'(x)}$
Here, $\lim\limits_{x \to \infty} f(x)=(\dfrac{\ln 3}{\ln 2}) \lim\limits_{x \to \infty}\dfrac{\ln x}{\ln (x+3)}=\dfrac{\infty}{\infty}$
This shows an indeterminate form of a limit, so we need to use L'Hospital's rule.
$(\dfrac{\ln 3}{\ln 2}) \lim\limits_{x \to \infty} \dfrac{1/x}{1/x+3}=(\dfrac{\ln 3}{\ln 2}) \lim\limits_{x \to \infty} \dfrac{x+3}{x}=\dfrac{\infty}{\infty}$
Again apply L'Hospital's rule.
$(\dfrac{\ln 3}{\ln 2}) \lim\limits_{x \to \infty} \dfrac{1}{1}=(\dfrac{\ln 3}{\ln 2})$
