Question

The value of $\underset{h\to 0}{lim}\left(\frac{(x+h{)}^{1/3}-x^{1/3}}{h}\right)$ equals to

• A. $\frac{1}{3}{x}^{2/3}$
• B. $\frac{1}{3}{x}^{-2/3}$
• C. $\frac{1}{3}{x}^{1/3}$
• D. $3x^{-2/3}$

Answer 1

Answer: (B)

$\frac{1}{3}{x}^{-2/3}$

$\underset{h\to 0}{lim}\left(\frac{(x+h{)}^{1/3}-x^{1/3}}{h}\right),\frac{0}{0}$ form
Thus using L Hospital,
Required limit is,
$=\underset{h\to 0}{lim}\left(\frac{1/3(x+h{)}^{-2/3}}{1}\right)=\frac{1}{3}{x}^{-2/3}$
Hence, option 'B' is correct.