Compute the following limit:

http://math.webwork.rochester.edu:80...2844a6b771.png =

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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- Dec 9th 2009, 03:54 PMtwirlonwaterLimit Problem with Fundamental Theorem of Calculus
Compute the following limit:

http://math.webwork.rochester.edu:80...2844a6b771.png =

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. - Dec 9th 2009, 04:09 PMScott H
Hint: let $\displaystyle F$ be an antiderivative of $\displaystyle \sqrt[3]{512-3t^2}$. Then

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

Have you learned L'Hospital's rule yet? - Dec 9th 2009, 04:19 PMtwirlonwater
Thank you for your help.

Yes, we did learn that. I used it and somehow got 0 for an answer, and that was wrong. - Dec 9th 2009, 05:28 PMScott H
Here is another hint. L'Hospital's rule states that

$\displaystyle \lim_{x\rightarrow 0}\frac{x}{F(x^2)-F(x)}=\lim_{x\rightarrow 0}\frac{1}{2xf(x^2)-f(x)}.$ - Dec 9th 2009, 05:53 PMProve It
- Dec 10th 2009, 04:54 AMHallsofIvy
That's incorrect. The derivative of $\displaystyle \int_x^{x^2} F(t) dt$ is $\displaystyle 2xF(x^2)- F(x)$. Use the chain rule with $\displaystyle u= x^2$

$\displaystyle \frac{d}{dx}\int_a^{x^2} F(t)dt= \frac{du}{dx}\frac{d}{du}\int_a^u F(t)dt= (2x)F(u)= 2xF(x^2)$. - Dec 10th 2009, 08:42 AMtom@ballooncalculus
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

and must have thought it was trying and failing to say something like

which Scott clearly did say in his other post further down. (OK, next time I'll skip the psychologicals and just correct the error.) - Dec 10th 2009, 12:18 PMScott H
What post of mine did he confuse this one with?

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