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Math Help - Limit Problem with Fundamental Theorem of Calculus

  1. #1
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    Limit Problem with Fundamental Theorem of Calculus

    Compute the following limit:

    =


    I tried to do this problem in a variety of ways, yet I always got the wrong answer. Any help would be absolutely appreciated. Thank you.
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  2. #2
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    Hint: let F be an antiderivative of \sqrt[3]{512-3t^2}. Then

    \lim_{x\rightarrow 0}\frac{x}{\int_x^{x^2}\sqrt[3]{512-3t^2}\,dt}=\lim_{x\rightarrow 0}\frac{x}{F(x^2)-F(x)}.

    Have you learned L'Hospital's rule yet?
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  3. #3
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    Thank you for your help.

    Yes, we did learn that. I used it and somehow got 0 for an answer, and that was wrong.
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  4. #4
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    Here is another hint. L'Hospital's rule states that

    \lim_{x\rightarrow 0}\frac{x}{F(x^2)-F(x)}=\lim_{x\rightarrow 0}\frac{1}{2xf(x^2)-f(x)}.
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  5. #5
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    Quote Originally Posted by Scott H View Post
    Here is another hint. L'Hospital's rule states that

    \lim_{x\rightarrow 0}\frac{x}{F(x^2)-F(x)}=\lim_{x\rightarrow 0}\frac{1}{2xf(x^2)-f(x)}.
    This is, of course, assuming that the denominator also tends to 0...
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  6. #6
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    Quote Originally Posted by Scott H View Post
    Hint: let F be an antiderivative of \sqrt[3]{512-3t^2}. Then

    \lim_{x\rightarrow 0}\frac{x}{\int_x^{x^2}\sqrt[3]{512-3t^2}\,dt}=\lim_{x\rightarrow 0}\frac{x}{F(x^2)-F(x)}.

    Have you learned L'Hospital's rule yet?
    That's incorrect. The derivative of \int_x^{x^2} F(t) dt is 2xF(x^2)- F(x). Use the chain rule with u= x^2
    \frac{d}{dx}\int_a^{x^2} F(t)dt= \frac{du}{dx}\frac{d}{du}\int_a^u F(t)dt= (2x)F(u)= 2xF(x^2).
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  7. #7
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    Just to note - all here are really in agreement, but H(I) has (unless it's me) confused two of Scott H's posts.

    Edit: sorry, I didn't help, did I! I just meant that H(I) mis-read

    Quote Originally Posted by Scott H View Post
    Hint: let F be an antiderivative of \sqrt[3]{512-3t^2}. Then



    \lim_{x\rightarrow 0}\frac{x}{\int_x^{x^2}\sqrt[3]{512-3t^2}\,dt}=\lim_{x\rightarrow 0}\frac{x}{F(x^2)-F(x)}.



    Have you learned L'Hospital's rule yet?
    and must have thought it was trying and failing to say something like

    Quote Originally Posted by Scott H View Post
    Here is another hint. L'Hospital's rule states that



    \lim_{x\rightarrow 0}\frac{x}{F(x^2)-F(x)}=\lim_{x\rightarrow 0}\frac{1}{2xf(x^2)-f(x)}.
    which Scott clearly did say in his other post further down. (OK, next time I'll skip the psychologicals and just correct the error.)
    Last edited by tom@ballooncalculus; December 10th 2009 at 02:30 PM.
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  8. #8
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    What post of mine did he confuse this one with?

    My post here is correct, as we have defined F to be an antiderivative of \sqrt[3]{512-3t^2} rather than \sqrt[3]{512-3t^2} itself.
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