Question

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

Answer 1

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

$\Rightarrow {lim}_{x\to 3}\frac{(x{)}^{\frac{1}{4}}-(3{)}^{\frac{1}{4}}}{(x{)}^{\frac{1}{3}}-(3{)}^{\frac{1}{3}}}$

Apply L-Hospital rule,

$={lim}_{x\to 3}\frac{\frac{1}{4}(x{)}^{\frac{1}{4}-1}-0}{\frac{1}{3}(x{)}^{\frac{1}{3}-1}-0}$

$={lim}_{x\to 3}\frac{3}{4}\frac{x^{-\frac{3}{4}}}{x^{-\frac{2}{3}}}$

$={lim}_{x\to 3}\frac{3}{4}{x}^{\frac{2}{3}-\frac{3}{4}}$

$={lim}_{x\to 3}\frac{3}{4}{x}^{-\frac{1}{12}}$

$=\frac{3}{4}\times (3{)}^{-\frac{1}{12}}$

$=\frac{1}{4}{x}^{\frac{11}{12}}$