Evaluate the following limit : lim(x→1) (1 - x^-1/3)/(1 - x^-2/3)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to 1}\cfrac{1-\text x^{-1/3}}{1-\text x^{-2/3}} \)

lim(x→1) (1 - x^-1/3)/(1 - x^-2/3)

Answer 1

Since the form is indeterminant

\(=\cfrac00\)

Method 1: factorization

Since a³ - b³ = (a + b) (a² + b² + ab) & a² - b² = (a + b) (a - b)

Method 2: By L hospital rule:

Differentiating numerator and denominator separately: