Question

$\underset{x\to 64}{lim}\frac{x^{\frac{1}{6}}-2}{x^{\frac{1}{3}}-4}$

Answer 1

$\underset{x\to 64}{lim}\frac{x^{\frac{1}{6}}-2}{x^{\frac{1}{3}}-4}$
$lety=x^{\frac{1}{6}}$

$whenx\to 64,y\to 2$

${lim}_{x\to 64}\frac{x^{\frac{1}{6}}-2}{x^{\frac{1}{3}}-4}=\underset{y\to 2}{lim}\frac{y-2}{y^2-4}$
$\Rightarrow \underset{y\to 2}{lim}\frac{y-2}{y^2-4}\left\{\frac{0}{0}form\right\}$

Using L-Hopital rule.

$\Rightarrow \underset{y\to 2}{lim}\frac{1}{2y}\Rightarrow \frac{1}{4}$