Question

The value of $f\left(0\right)$, so that the 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. $a^{3/2}$
• B. $a^{1/2}$
• C. $-a^{1/2}$
• D. $-a^{3/2}$

Answer 1

The correct option is C $-a^{1/2}$

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

$f\left(x\right)$ is continuous for all $x$ so it is continuous at $x=0$

$\therefore f\left(0\right)={lim}_{x\to 0}f\left(x\right)$

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

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

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

$=\frac{-a\times 2\sqrt{a}}{2a}=-\sqrt{a}$