lim-x-1-x-1-1-x-1-3-1-x-2-1-3-

Question Number 91133 by john santu last updated on 28/Apr/20

\lim_{x \to 1} \frac{(x - 1) + \sqrt[3]{1 - x}}{\sqrt[3]{1 - x^{2}}} =

Commented by john santu last updated on 28/Apr/20

\lim_{x \to 1} \frac{(x - 1) + (1 - \frac{x}{3})}{\sqrt[3]{2} (1 - \frac{x}{3})} =

\frac{1}{\sqrt[3]{2}} \times \lim_{x \to 1} (\frac{\frac{2}{3} x}{\frac{2}{3} x}) = \frac{1}{\sqrt[3]{2}} .

Commented by mathmax by abdo last updated on 28/Apr/20

l e t f (x) = \frac{(x - 1) +^{3} \sqrt{1 - x}}{(^{3} \sqrt{1 - x^{2}})} c h a n g e m e n t^{3} \sqrt{1 - x} = t g i v e

1 - x = t^{3} \Rightarrow f (x) = g (t) = \frac{- t^{3} + t}{t (^{3} \sqrt{1 + 1 - t^{3})}} = \frac{1 - t^{2}}{{(2 - t^{3})}^{\frac{1}{3}}}

= \frac{1 - t^{2}}{(^{3} \sqrt{2}) {(1 - \frac{t^{3}}{2})}^{\frac{1}{3}}} (x \to 1 \Rightarrow t \to 0) \Rightarrow g (t) \sim \frac{1 - t^{2}}{(^{3} \sqrt{2}) (1 - \frac{1}{6} t^{3})} \Rightarrow

{l i m}_{t \to 0} g (t) = \frac{1}{(^{3} \sqrt{2})} = {l i m}_{x \to 1} f (x)

Answered by john santu last updated on 28/Apr/20

\lim_{x \to 1} \frac{\sqrt[3]{x - 1} (\sqrt[3]{{(x - 1)}^{2}} - 1)}{\sqrt[3]{1 - x} \sqrt[3]{1 + x}} =

\lim_{x \to 1} \frac{1 - \sqrt[3]{{(x - 1)}^{2}}}{\sqrt[3]{1 + x}} = \frac{1}{\sqrt[3]{2}}