Question

$\underset{x\to 0}{lim}\frac{(27+x{)}^{\frac{1}{3}}-3}{9-(27+x{)}^{\frac{2}{3}}}\text{equals}$ :

• A. $-\frac{1}{6}$
• B. $\frac{1}{6}$
• C. $\frac{1}{3}$
• D. $-\frac{1}{3}$

Answer 1

The correct option is A $-\frac{1}{6}$

$\underset{x\to 0}{lim}\frac{(27+x{)}^{\frac{1}{3}}-3}{9-(27+x{)}^{\frac{2}{3}}}$
By using L'Hopitals rule, as it is in the form $\frac{0}{0}$ we get-
$\underset{x\to 0}{lim},-\left[\frac{1}{2}\frac{(27+x{)}^{\frac{-2}{3}}}{(27+x{)}^{\frac{-1}{3}}}\right]=-\frac{1}{6}$