Question

${lim}_{x\to 1}\frac{1-x^{-1/3}}{1-x^{-2/3}}$

Answer 1

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

Here putting the limit $x\to 1$ we

get $\frac{0}{0}$ from (undefined form )

None we can apply L-Hospital Rule

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

$\Rightarrow \underset{x\to 1}{lim}\frac{0-\left(\frac{-1}{3}{x}^{-1/3-1}\right)}{0-(\frac{-2}{3}{x}^{-2/3}-1)}$ $\left\{\frac{d}{dx}\right(x^n)=n{x}^{n-1}=d(x{)}^n$

$\Rightarrow \underset{x\to 1}{lim}\frac{+\frac{1}{3}{x}^{-4/3}}{+\frac{2}{3}{x}^{-5/3}}$

$\Rightarrow \underset{x\to 1}{lim}\frac{1}{2}{x}^{-4/3}+\frac{5}{3}$

$\Rightarrow \underset{x\to 1}{lim}\frac{1}{2}x\frac{1}{3}$

$\Rightarrow \frac{1}{2}(1{)}^{1/3}\Rightarrow \frac{1}{2}$