Question

If $f\left(x\right)=\underset{n\to \infty }{lim}n(x^{\frac{1}{n}}-1)$, then for $x>0,y>0,f\left(xy\right)$ is equal to

• A. $f\left(x\right)f\left(y\right)$
• B. $f\left(x\right)+f\left(y\right)$
• C. $f\left(x\right)-f\left(y\right)$
• D. none of these

Answer 1

Answer: (B)

$f\left(x\right)+f\left(y\right)$

$f\left(x\right)=\underset{n\to \infty }{lim}n(x^{\frac{1}{n}}-1)$

$=\underset{n\to \infty }{lim}\frac{x^{\frac{1}{n}}-1}{\frac{1}{n}}$

$=\underset{m\to 0}{lim}\frac{x^m-1}{m}\text{(where}\frac{1}{n}\text{is replaced by m)}$

$=\ln x$

$\ln \left(xy\right)=\ln x+\ln y$

$\Rightarrow f\left(xy\right)=f\left(x\right)+f\left(y\right)$