Question

Evaluate : $\underset{x\to 3}{lim}\frac{\sqrt[4]{4x}-\sqrt[4]{43}}{\sqrt[3]{3x}-\sqrt[3]{33}}$

Answer 1

${lim}_{x\to 3}\frac{\sqrt[4]{4x}-\sqrt[4]{43}}{\sqrt[3]{3x}-\sqrt[3]{33}}={lim}_{x\to 3}\frac{{\left(x\right)}^{1/4}-3^{1/4}}{{\left(x\right)}^{1/3}-{\left(3\right)}^{1/3}}$

Apply L-hospital rule,

$={lim}_{x\to 3}\frac{\frac{1}{4}{\left(x\right)}^{-3/4}}{\frac{1}{3}{\left(x\right)}^{-2/3}}=\frac{3}{4}{x}^{2/3-3/4}={lim}_{x\to 3}{x}^{-1/12}=\frac{3}{4}{\left(3\right)}^{-1/12}=\frac{{\left(3\right)}^{11/12}}{4}$

Hence, this is the answer.