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.
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?
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.)