Mathematics - Limits Question with Solution | TestHub

Given limit is$\underset{x\to 0}{\lim }\frac{{\left(27+x\right)}^{\frac{1}{3}}-3 }{9-{\left(27+x\right)}^{\frac{2}{3}}} \left(\frac{0}{0} \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\right)$

Applying L'Hospital's rule,

$\Rightarrow \underset{x\to 0 }{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\frac{1}{3}{\left(27+x\right)}^{-\frac{2}{3}} }{-\frac{2}{3}{\left(27+x\right)}^{-\frac{1}{3}}}=\frac{{\left(27\right)}^{-\frac{2}{3}}}{-2{\left(27\right)}^{-\frac{1}{3}}}$

$=\frac{-1}{2{\left(27\right)}^{\frac{1}{3}} }=-\frac{1}{6}$.