# Thread: Limit Problem with Fundamental Theorem of Calculus

1. ## 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.

2. 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?

3. Thank you for your help.

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

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

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

5. 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...

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

7. 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.)

8. 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.