Question

Evaluate the limit:$\underset{x\to 1}{lim}\frac{1-x^{-1/3}}{1-x^{-2/3}}$

Answer 1

Given: $\underset{x\to 1}{lim}\frac{1-x^{-1/3}}{1-x^{-2/3}}becomes\frac{0}{0}form$

Using Factorization method

$\Rightarrow \underset{x\to 1}{lim}\frac{1-x^{-1/3}}{1-x^{-2/3}}=\underset{x\to 1}{lim}\frac{1-x^{-1/3}}{1^2-{\left(x^{-1/3}\right)}^2}$

$[\because a^2-b^2=(a+b\left)\right(a-b\left)\right]$

$\Rightarrow \underset{x\to 1}{lim}\frac{1-x^{-1/3}}{(1+x^{-1/3})(1-x^{-1/3})}=\underset{x\to 1}{lim}\frac{1}{1+x^{-1/3}}$

$=\frac{1}{1+{\left(1\right)}^{-1/3}}=\frac{1}{2}$

Therefore, $\underset{x\to 1}{lim}\frac{1-x^{-1/3}}{1-x^{-2/3}}=\frac{1}{2}$