Question

Evaluate

$li{m}_{x\to 0}\frac{\sqrt[K]{K1+x}-1}{x}$ ( K is a positive integer )

• A. K
• B. -K
• C. $\frac{1}{K}$
• D. $-\frac{1}{K}$

Answer 1

Answer: (C)

$\frac{1}{K}$

$\underset{x\to 0}{lim}$ $\frac{\sqrt[K]{K1+x}-1}{x}$ ($\frac{0}{0}$ form)

$=\underset{x\to 0}{lt}$ $\frac{\frac{1}{K}{(1+x)}^{K-1}}{1}$ (Using $L$ Hospitals rule)

$=\frac{1}{K}{(1+0)}^{K-1}$

$=\frac{1}{K}{\left(1\right)}^{K-1}$

$=\frac{1}{K}$