Question

The value of $f\left(0\right)$, so that function, $f\left(x\right)=\frac{\sqrt{a^2-ax+x^2}-\sqrt{a^2+ax+x^2}}{\sqrt{a+x}-\sqrt{a-x}}$ becomes continuous for all $x$, is given by-

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

Answer 1

Answer: (D)

$-a\sqrt{a}$

$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}\frac{\sqrt{a^2-ax+x^2}-\sqrt{a^2+ax+x^2}}{\sqrt{a+x}-\sqrt{a-x}}$
On rationalizing both Nr. and Dr. we get
$\underset{x\to 0}{lim}f\left(x\right)=-a\sqrt{a}$ so $f\left(0\right)=-a\sqrt{a}$