Question

$\begin{array}{c}lim\\ x\to 0\end{array}\frac{{(1+x)}^{1/6}-{(1-x)}^{1/6}}{x}$

Answer 1

Given,

${lim}_{x\to 0}\frac{{(1+x)}^{\frac{1}{6}}-{(1-x)}^{\frac{1}{6}}}{x}$

Apply L-hospital rule,

$={lim}_{x\to 0}\frac{\frac{d}{dx}[{(1+x)}^{\frac{1}{6}}-{(1-x)}^{\frac{1}{6}}]}{\frac{d}{dx}\left[x\right]}$

$={lim}_{x\to 0}\frac{\frac{1}{6{(1+x)}^{\frac{5}{6}}}+\frac{1}{6{(1-x)}^{\frac{5}{6}}}}{1}$

$={lim}_{x\to 0}\frac{1}{6{(1+x)}^{\frac{5}{6}}}+\frac{1}{6{(1-x)}^{\frac{5}{6}}}$

$=\frac{1}{6{(1+0)}^{\frac{5}{6}}}+\frac{1}{6{(1-0)}^{\frac{5}{6}}}$

$=\frac{1}{6}+\frac{1}{6}$

$=\frac{1}{3}$

Hence, this is the answer.