Without l'Hopital's rule you can use the fact that $\displaystyle \lim_{x \to 0} \frac{\sin(x)}{x} = 1$
We have
$\displaystyle \lim_{u \to 0} \frac{\sin u^2}{u} = \lim_{u \to 0} \frac{\sin u^2}{u} \frac{u}{u} = \lim_{u \to 0} \frac{\sin u^2}{u^2} \lim_{u \to 0} u$
If $\displaystyle u \to 0$ then $\displaystyle u^2 \to 0$ thus $\displaystyle \lim_{u \to 0} \frac{\sin u^2}{u^2} = 1$ and $\displaystyle \lim_{u \to 0} u = 0$ therefore
$\displaystyle \lim_{u \to 0} \frac{\sin u^2}{u} = 0$