Limit Problem with Fundamental Theorem of Calculus

**twirlonwater** · December 9th 2009, 03:54 PM

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.

**Scott H** · December 9th 2009, 04:09 PM

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?

**twirlonwater** · December 9th 2009, 04:19 PM

Thank you for your help.

Yes, we did learn that. I used it and somehow got 0 for an answer, and that was wrong.

**Scott H** · December 9th 2009, 05:28 PM

Here is another hint. L'Hospital's rule states that

$\lim_{x\rightarrow 0}\frac{x}{F(x^2)-F(x)}=\lim_{x\rightarrow 0}\frac{1}{2xf(x^2)-f(x)}.$

**Prove It** · December 9th 2009, 05:53 PM

Originally Posted by Scott H

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...

**HallsofIvy** · December 10th 2009, 04:54 AM

Originally Posted by Scott H

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 $u= x^2$
$\frac{d}{dx}\int_a^{x^2} F(t)dt= \frac{du}{dx}\frac{d}{du}\int_a^u F(t)dt= (2x)F(u)= 2xF(x^2)$ .

**tom@ballooncalculus** · December 10th 2009, 08:42 AM

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

Originally Posted by Scott H

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

Originally Posted by Scott H

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.)

**Scott H** · December 10th 2009, 12:18 PM

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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