Question

$\underset{x\to 0}{lim}\frac{\sqrt[3]{3(1+\sin x)}-\sqrt[3]{3(1-\sin x)}}{x}=\frac{2}{b}$. Find b

Answer 1

Let $L=\underset{x\to 0}{lim}\frac{\sqrt[3]{3(1+\sin x)}-\sqrt[3]{3(1-\sin x)}}{x}$
Now let $a=\sqrt[3]{3(1+\sin x)}$ and $b=\sqrt[3]{3(1-\sin x)}$
Thus using, $a^3-b^3=(a-b)(a^2+b^2+ab)$
$L=\underset{x\to 0}{lim}\frac{(1+\sin x)-(1-\sin x)}{x(a^2+ab+b^2)}$
$=\underset{x\to 0}{lim}\frac{2\frac{\sin x}{x}}{(a^2+ab+b^2)}=\frac{2}{3}$
$\Rightarrow b=3$