# Math Help - how to evaluate this algebraically

1. ## how to evaluate this algebraically

I know this goes to zero as u goes to zero from my calculator, but how does one break this down algebraically?

2. ## Re: how to evaluate this algebraically

L'Hôpital's rule:

$\lim_{u \to 0} \frac{\sin u^2}{u} = \lim_{u \to 0} \frac{2u \cos u^2}{1} = 0$

3. ## Re: how to evaluate this algebraically

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

If $u \to 0$ then $u^2 \to 0$ thus $\lim_{u \to 0} \frac{\sin u^2}{u^2} = 1$ and $\lim_{u \to 0} u = 0$ therefore
$\lim_{u \to 0} \frac{\sin u^2}{u} = 0$