Question

The value of $\underset{x\to 0}{lim}\frac{(1+x{)}^{\frac{1}{3}}-(1-x{)}^{\frac{1}{3}}}{x}$ is ?

• A. $\frac{2}{3}$
• B. $\frac{1}{3}$
• C. $1$
• D. $\frac{5}{3}$

Answer 1

The correct option is A $\frac{2}{3}$

$L=\underset{x\to 0}{lim}\frac{(1+x{)}^{1/3}-(1-x{)}^{1/3}}{x}$
Using binomial expansion: $(1+x{)}^n=1+nx+o(x^2)$
$\therefore L=\underset{x\to 0}{lim}\frac{1+\frac{1}{3}x+o\left(x^2\right)-1+\frac{1}{3}x+o\left(x^2\right)}{x}$
$=\underset{x\to 0}{lim}\frac{\frac{2}{3}x+o\left(x^2\right)}{x}$
$=\underset{x\to 0}{lim}\frac{2}{3}+o\left(x\right)$
$=\frac{2}{3}$
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Here $o\left(x^n\right)$ is used to denote a power series where the smallest power of $x$ is $n$.