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Math Help - limit as x goes to zero of sin(x^2)/x

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    limit as x goes to zero of sin(x^2)/x

    limit as x goes to zero of sin(x^2)/x-screen-shot-2012-10-24-12.09.46-am.png

    Does this require l'Hospitals's Rule or is there another way to do this?

    l'hospital's rule would give lim x->0 cos(x^2)/1 which is = 1/1=1
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    Re: limit as x goes to zero of sin(x^2)/x

    You are forgetting to apply the chain rule in the numerator...you could also multiply by x/x to get the same result of 0.
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    Re: limit as x goes to zero of sin(x^2)/x

    so for the the above it should be cos(x^2)*2x/1 which would be 1*0/1=0

    multiplying by x/x we would get xsin(x^2)/x^2 which is equal is also equal to zero too but how? I feel like i'm missing some obvious step for version where we multiply by x/x

    also is sin(x^2)/x=sin^2x/x=(sinx)^2/x ? or is it just the last two that are the same?
    Last edited by kingsolomonsgrave; October 23rd 2012 at 09:31 PM.
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    Re: limit as x goes to zero of sin(x^2)/x

    Quote Originally Posted by kingsolomonsgrave View Post
    so for the the above it should be cos(x^2)*2x/1 which would be 1*0/1=0

    multiplying by x/x we would get xsin(x^2)/x^2 which is equal is also equal to zero too but how? I feel like i'm missing some obvious step for version where we multiply by x/x

    also is sin(x^2)/x=sin^2x/x=(sinx)^2/x ? or is it just the last two that are the same.]?
    \displaystyle \begin{align*} \lim_{x \to 0}\frac{\sin{\left(x^2\right)}}{x} &= \lim_{x \to 0}\frac{x\sin{\left(x^2\right)}}{x^2} \\ &= \lim_{x \to 0} x \cdot \lim_{x \to 0}\frac{\sin{\left(x^2\right)}}{x^2} \\ &= 0 \cdot 1 \\ &= 0 \end{align*}


    Also, no, \displaystyle \begin{align*} \sin{\left(x^2\right)} \neq \sin^2{x} \end{align*}, rather \displaystyle \begin{align*} \sin^2{x} = \left(\sin{x}\right)^2 \end{align*}.
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    Re: limit as x goes to zero of sin(x^2)/x

    We have:

    \lim_{x\to0}\frac{x\sin(x^2)}{x^2}=\lim_{x\to0}x \cdot \lim_{x\to0}\frac{\sin(x^2)}{x^2}

    For the second limit let u=x^2 and as x\to0 we have u\to0 so we may write:

    \lim_{x\to0}x\cdot\lim_{u\to0}\frac{\sin(u)}{u}=0 \cdot1=0

    I am assuming you have been given \lim_{u\to0}\frac{\sin(u)}{u}=1 to use for such limits.
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    Re: limit as x goes to zero of sin(x^2)/x

    Thanks, yes we were given lim as x-> 0 sin(u)/u = 1

    thanks much
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