Answer
$\dfrac{\ln 3}{\ln 2}$
Work Step by Step
Here,we have $\lim\limits_{x \to \infty} f(x)=(\dfrac{\ln 3}{\ln 2}) \lim\limits_{x \to \infty}\dfrac{\ln x}{\ln (x+3)}$
and $\lim\limits_{x \to \infty} f(\infty)=\dfrac{\infty}{\infty}$
This shows an indeterminate form of limit , thus we will apply L-Hospital's rule such as:
$\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{p'(x)}{q'(x)}$
$(\dfrac{\ln 3}{\ln 2}) \lim\limits_{x \to \infty} \dfrac{1/x}{1/(x+3)}=\dfrac{\ln 3}{\ln 2} [\lim\limits_{x \to \infty} \dfrac{x+3}{x}]=\dfrac{\infty}{\infty}$
Now, again apply L-Hospital's rule.
$\dfrac{\ln 3}{\ln 2} \lim\limits_{x \to \infty} (\dfrac{1}{1})=\dfrac{\ln 3}{\ln 2}$
