Question

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

• A. $a^{3 / 2}$
• B. $a^{1 / 2}$
• C. $-a^{1 / 2}$
• D. $-a^{3 / 2}$

Answer 1

Answer: (C)

$f(x)=\frac{\sqrt{a^2-a x+x^2}-\sqrt{a^2+a x+x^2}}{\sqrt{a+x}-\sqrt{a-x}}$
$\times \frac{\sqrt{a^2-a x+x^2}+\sqrt{a^2+a x+x^2}}{\sqrt{a^2-a x+x^2}+\sqrt{a^2+a x+x^2}} \times \frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}+\sqrt{a-x}}$
$\Rightarrow \displaystyle \lim _{x \rightarrow 0} f(x)$
$=\displaystyle\lim _{x \rightarrow 0} \frac{-2 a x(\sqrt{a+x}+\sqrt{a-x})}{2 x\left(\sqrt{a^2-a x+x^2}+\sqrt{a^2+a x+x^2}\right)}$
$=\frac{-a(2 \sqrt{a})}{a+a}=-\sqrt{a}$