$\underset{x \to 1}{\lim}\left( \frac{x^\frac{1}{3} -1}{x^\frac{1}{4} -1}\right)$
Applying L- Hospital' rule and Differentiating the numerator and denominator ,
$\underset{x \to 1}{\lim}\left( \frac{\frac{1}{3}x^\frac{-2}{3} }{\frac{1}{4}x^\frac{-3}{4} }\right )$
Applying the limit we get $\dfrac{4}{3}$ as answer.