Question

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

Answer 1

Answer: 3

$\underset{x\to 2}{\lim }\frac{(10-x{)}^{1/3}-2}{x-2}$
$=\underset{h\to 0}{\lim }\frac{(8-h{)}^{1/3}-2}{h}$
$($ Put $x=2+h)$
$=\underset{h\to 0}{\lim }\frac{2{\left(1-\frac{h}{8}\right)}^{1/3}-2}{h}$
$=2\underset{h\to 0}{\lim }\frac{{\left(1-\frac{h}{8}\right)}^{1/3}-1}{h}$
$=2\underset{h\to 0}{\lim }\frac{1-\frac{1}{3}\frac{h}{8}-1}{h}$
$=-\frac{1}{12}$