Question

$\underset{x\to 0}{lim}f\left(x\right)$, where $f\left(x\right)=\frac{\sqrt{a^2-ax+x^2}-\sqrt{a^2+ax+x^2}}{\sqrt{a+x}-\sqrt{a-x}}$ is

• A. $a$
• B. $-a$
• C. $\sqrt{a}$
• D. $-\sqrt{a}$

Answer 1

The correct option is D $-\sqrt{a}$

$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}\frac{\sqrt{a^2-ax+x^2}-\sqrt{a^2+ax+x^2}}{\sqrt{a+x}-\sqrt{a-x}}$

Rationalizing the numerator and denominator

$=\underset{x\to 0}{lim}\frac{-2ax}{\sqrt{a^2-ax+x^2}+\sqrt{a^2+ax+x^2}}\times \frac{\sqrt{a+x}+\sqrt{a-x}}{2x}$

$=\underset{x\to 0}{lim}\frac{-a(\sqrt{a+x}+\sqrt{a-x})}{\sqrt{a^2-ax+x^2}+\sqrt{a^2+ax+x^2}}$

$=\frac{-a.2\sqrt{a}}{2a}=-\sqrt{a}$
Hence, option 'D' is correct.