Question

Value of limit $\underset{x\to 0^{+}}{lim}\frac{a^{\sqrt{x}}-a^{1/\sqrt{x}}}{a^{\sqrt{x}}+a^{1/\sqrt{x}}},a>1,$ is

• A. $4$
• B. $2$
• C. $-1$
• D. $0$

Answer 1

The correct option is C $-1$

$\underset{x\to 0}{lim}\frac{a^{\sqrt{x}}-a^{1/\sqrt{x}}}{a^{\sqrt{x}}+a^{1/\sqrt{x}}},a>1$
The domain of above expression is $x\in R^{+}$
$=\underset{x\to 0^{+}}{lim}\frac{a^{\sqrt{x}-1/\sqrt{x}}-1}{a^{\sqrt{x}-1/\sqrt{x}}+1}=\frac{a^{-\infty }-1}{a^{-\infty }+1}=-1(a>1\Rightarrow a^{-\infty }\to 0)$