Answer
The l'Hospital's conditions are not fulfilled
Work Step by Step
The l'Hospital's rule is used when the limit has an indeterminte form type $\frac{0}{0}$ or $\frac{\infty}{\infty}$
The given limit is:
$$\lim\limits_{x \to 0}\frac{x^{2}}{x^{2}+3}$$
So using the direct substitution of limit property since the function inside the limit is continuous at $x=0$:
$$\lim\limits_{x \to 0}\frac{x^{2}}{x^{2}+3}=\lim\limits_{x \to 0}\frac{0^{2}}{0^{2}+3}=\lim\limits_{x \to 0}\frac{0}{3}=0$$
Since the limit does not have an indeterminate form type $\frac{0}{0}$ or $\frac{\infty}{\infty}$ and exactly is equal to $0$ by the direct substitution of limit property so the l'Hospital's should not be used.