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Math Help - limits help please!!!

  1. #1
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    limits help please!!!

    i don't know how to prove the following limits

    limit of ( 1 - cos x) / x^2 when x is +infinity

    limit of (tan4x) / 6 when x is 0

    limit of (sin x^2) / x

    in the first equation i got to the point of multiplying by the reciprocal but i don't know what to do next

    in the second equation i converted tan to sin/cos but i dont know where to go from there
    Last edited by ubhutto; October 1st 2012 at 05:04 PM.
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  2. #2
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    Re: limits help please!!!

    Quote Originally Posted by ubhutto View Post
    i don't know how to prove the following limits

    limit of ( 1 - cos x) / x^2 when x is +infinity

    limit of (tan4x) / 6 when x is 0

    limit of (sin x^2) / x

    in the first equation i got to the point of multiplying by the reciprocal but i don't know what to do next

    in the second equation i converted tan to sin/cos but i dont know where to go from there
    In the first, \displaystyle \begin{align*} 1 - \cos{x} \end{align*} is bounded between 0 and 2, and the denominator goes to \displaystyle \begin{align*} \infty \end{align*}, which makes the fraction go to 0.

    For the second, surely you can just substitute in x = 0.

    For the third, what is the value of x going to?
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  3. #3
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    Re: limits help please!!!

    it is going to 0
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    Re: limits help please!!!

    Well multiply top and bottom by x to give

    \displaystyle \begin{align*} \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} \end{align*}

    Can you go from there?
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  5. #5
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    Re: limits help please!!!

    i dont understand what to do with the sinx^2 / x^2 , how do i simplify it, if you can help me with that part ill be able to do the rest of the problems i have
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    Re: limits help please!!!

    Quote Originally Posted by ubhutto View Post
    i dont understand what to do with the sinx^2 / x^2 , how do i simplify it, if you can help me with that part ill be able to do the rest of the problems i have
    You should know that \displaystyle \begin{align*} \lim_{X \to 0}\frac{\sin{X}}{X} = 1 \end{align*}. Here your \displaystyle \begin{align*} X \end{align*} just happens to be \displaystyle \begin{align*}  x^2 \end{align*}.
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