Question

$\underset{x\to \infty }{lim}x(a^{\frac{1}{x}}-b^{\frac{1}{x}})=$

• A. $1$
• B. $lo{g}_{e}a/b$
• C. $lo{g}_e\left(ab\right)$
• D. $0$

Answer 1

The correct option is B $lo{g}_{e}a/b$

Now,

$\underset{x\to \infty }{lim}x(a^{\frac{1}{x}}-b^{\frac{1}{x}})$.......(1)

Put $y=\frac{1}{x}$ then (1) becomes

$=\underset{y\to 0}{lim}\frac{(a^y-b^y)}{y}$

$=\underset{y\to 0}{lim}\frac{\left(\right\{a^y-1\}-\{b^y-1\left\}\right)}{y}$

$={\log }_{e}a-{\log }_{e}b$

$={\log }_e\frac{a}{b}$.