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. $\frac{f\left(x\right)}{f\left(y\right)}$

Answer 1

The correct option is B $f\left(x\right)+f\left(y\right)$

$f\left(x\right)=\underset{n\to \infty }{lim}n(x^{\frac{1}{n}}-1)$
$=\underset{n\to \infty }{lim}\frac{x^{\frac{1}{n}}-1}{\frac{1}{n}}$
Now, replace $\frac{1}{n}$ by $m,$ we get
$\Rightarrow \underset{m\to 0}{lim}\frac{x^m-1}{m}$
$={\log }_{e}x$

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