# how to evaluate this algebraically

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• October 23rd 2012, 03:00 PM
kingsolomonsgrave
how to evaluate this algebraically
Attachment 25372

I know this goes to zero as u goes to zero from my calculator, but how does one break this down algebraically?
• October 23rd 2012, 03:12 PM
richard1234
Re: how to evaluate this algebraically
L'Hôpital's rule:

$\lim_{u \to 0} \frac{\sin u^2}{u} = \lim_{u \to 0} \frac{2u \cos u^2}{1} = 0$
• October 24th 2012, 07:21 AM
Siron
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
$\lim_{u \to 0} \frac{\sin u^2}{u} = 0$