$\displaystyle \lim_{x \to \0}\frac{(3+x)^{\frac{4}{7}} - (3-x)^{\frac{4}{7}}}{x}$, which gives the indeterminate form $\displaystyle \frac{0}{0}$. Using l'Hospital' rule gives:

$\displaystyle \lim_{x \to \0}[\frac{4}{7}(3+x)^{-\frac{3}{7}} - \frac{4}{7}(3+x)^{-\frac{3}{7}}] = 0$

But, the book gives a different answer and I confirmed their answer by using a different method. What I want to know exactly is why l'Hospital's rule failed to work for this limit.