Answer
$0$
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, $\lim\limits_{x \to 0^{+}} f(x)=\lim\limits_{x \to 0^{+}} \dfrac{(\ln x)^2}{1/x^2}$
Now, $\lim\limits_{x \to 0^{+}} \dfrac{2 \ln x/x}{-1/x^2}=\dfrac{-\infty}{-\infty}$
This shows an indeterminate form of the limit, so we need to use L'Hospital's rule.
$\lim\limits_{x \to 0^{+}} \dfrac{2/ x}{1/x^2}=2(0)=0 $
