Question

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

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

Answer 1

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

$\underset{x\to 0}{lim}\frac{(27+x{)}^{\frac{1}{3}}-3}{9-(27+x{)}^{\frac{2}{3}}}\left(\frac{0}{0}\text{form}\right)$
Applying L Hosptial's rule
$=\underset{x\to 0}{lim}\left[\frac{\frac{1}{3}(27+x{)}^{\frac{-2}{3}}}{-\frac{2}{3}(27+x{)}^{\frac{-1}{3}}}\right]=-\frac{1}{6}$