Question

plz answer this question​

Answer 1

Step-by-step explanation:

We have,

$\lim _{x \rarr0 } \frac{ \sqrt[3]{1 + x} - \sqrt[3]{1 - x} }{x}$

$\lim _{x \rarr0 } \frac{( \sqrt[3]{1 + x} - \sqrt[3]{1 - x})((1 + x)^{ \frac{2}{3} } + (1 - {x}^{2}) ^{ \frac{2}{3} } + (1 - x)^{ \frac{2}{3} } ) }{x((1 + x)^{ \frac{2}{3} } + (1 - {x}^{2}) ^{ \frac{2}{3} } + (1 - x)^{ \frac{2}{3} } ) }$

$\lim _{x \rarr0 } \frac{( \sqrt[3]{1 + x} )^{3} - ( { \sqrt[3]{1 - x} })^{3} }{x((1 + x)^{ \frac{2}{3} } + (1 - {x}^{2} )^{ \frac{2}{3}} + (1 - x)^{ \frac{2}{3} } )}$

$\lim _{x \rarr0 } \frac{1 + x - 1 + x}{x((1 + x)^{ \frac{2}{3} } + (1 - {x}^{2} )^{ \frac{2}{3} } + (1 - x) ^ { \frac{2}{3} }) }$

$\lim _{x \rarr0 } \frac{2x}{x((1 + x)^{ \frac{2}{3} } + (1 - {x}^{2} ) ^{ \frac{2}{3} } + (1 - x)^{ \frac{2}{3} } )}$

$\lim _{x \rarr0 } \frac{2}{(1 + x) ^{ \frac{2}{3} } + (1 - x ^{2} )^{ \frac{2}{3} } + (1 - x) ^{ \frac{2}{3} } }$

$= \frac{2}{1 + 1 + 1}$

$= \frac{2}{3}$