The value of $\lim\limits_{x \rightarrow 0} \frac{(1+x)^{\frac{1}{3}}-(1-x)^{\frac{1}{3}}}{x}$ is:

2 / 3

$\lim\limits_{x \rightarrow 0} \frac{(1+x)^{\frac{1}{3}}-(1-x)^{\frac{1}{3}}}{x} \frac{(1+x)^{2 / 3}+(1-x)^{2 / 3}+(1+x)^{1 / 3}(1-x)^{1 / 3}}{(1+x)^{2 / 3}+(1-x)^{2 / 3}+(1+x)^{1 / 3}(1-x)^{1 / 3}}$

$=\lim\limits_{x \rightarrow 0} \frac{(1+x)-(1-x)}{(1+x)^{2 / 3}+(1-x)^{2 / 3}+(1+x)^{1 / 3}(1-x)^{1 / 3}} . \frac{1}{x}$

$=\lim\limits_{x \rightarrow 0} \frac{2}{(1+x)^{2 / 3}+(1-x)^{2 / 3}+(1+x)^{1 / 3}(1-x)^{1 / 3}}=\frac{2}{1+1+1}=\frac{2}{3}$

Hence (1) is the correct answer.