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Math Help - how to evaluate this algebraically

  1. #1
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    how to evaluate this algebraically

    how to evaluate this algebraically-screen-shot-2012-10-23-6.58.21-pm.png


    I know this goes to zero as u goes to zero from my calculator, but how does one break this down algebraically?
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  2. #2
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    Re: how to evaluate this algebraically

    L'H˘pital's rule:

    \lim_{u \to 0} \frac{\sin u^2}{u} = \lim_{u \to 0} \frac{2u \cos u^2}{1} = 0
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  3. #3
    MHF Contributor Siron's Avatar
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    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
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