Question

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

• A. $4$
• B. $2$
• C. $-1$
• D. $0$

Answer 1

The correct option is C $-1$

$\underset{x\to 0}{lim}\frac{a^{\sqrt{x}}-a^{\frac{1}{\sqrt{x}}}}{a^{\sqrt{x}}+a^{\frac{1}{\sqrt{x}}}}$ $a>1$
Substitute $x=t^2$.
Also as $x\to 0,t\to 0$
$\underset{t\to 0}{lim}\frac{a^t-a^{\frac{1}{t}}}{a^t+a^{\frac{1}{t}}}$
$or\underset{t\to 0}{lim}\frac{a^{t-\frac{1}{t}}-1}{a^{t-\frac{1}{t}}+1}$
$=\frac{a^{-\infty }-1}{a^{-\infty }+1}=\frac{0-1}{0+1}=-1$