$\underset{x\to 0}{lim} \frac{(27+x{)}^{1/3}-3}{9-(27+x{)}^{2/3}}$ is equal

Solution:

We have, $\underset{x\to 0}{lim} \frac{(27+x{)}^{1/3}-3}{9-(27+x{)}^{2/3}}$

$\begin{array}{l}=\underset{x\to 0}{lim} \frac{3\left[{\left(1+\frac{x}{27}\right)}^{1/3}-1\right]}{9\left[1-{\left(1+\frac{x}{27}\right)}^{2/3}\right]}=\underset{x\to 0}{lim} \frac{\left[\left(1+\frac{x}{3\times 27}+\dots \right)-1\right]}{3\left[1-\left(1+\frac{2x}{3\times 27}+\dots \right)\right]}\\ =\underset{x\to 0}{lim} \frac{1[x/81]}{3[-2x/81]}=-\frac{1}{6}\end{array}$