Question

If $\underset{x\to 2}{lim}\frac{(10-x{)}^{1/3}-2}{x-2}$, then the value of $\left|\frac{1}{\left(4L\right)}\right|$ is ............

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

$L=\underset{x\to 2}{lim}\frac{{(10-x)}^{1/3}-2}{x-2}=-\underset{x\to 2}{lim}\frac{{(10-x)}^{1/3}-8^{1/3}}{(10-x)-8}$

as $x\to 2;-x\to -2$

$10-x\to 8$

$=-\underset{(10-x)\to 8}{lim}\frac{{(10-x)}^{1/3}-8^{1/3}}{(10-x)-8}\left(\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n.a^{n-1}\right)$

$=\frac{1}{3}\cdot 8^{(\frac{1}{3}-1)}=\frac{1}{3}\times 8^{-2/3}=\frac{1}{12}=L$

$\frac{1}{4L}=\frac{1}{4\times \frac{1}{12}}=3$