Question

If $f\left(x\right)=\frac{\sqrt{a^2-ax+x^2}-\sqrt{a^2+ax+x^2}}{\sqrt{a+x}-\sqrt{a-x}}$ is continuous at $x=0$ then $f\left(0\right)=$

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

Answer 1

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

$\begin{array}{c}Lim\\ x\to 0\end{array}\left(\frac{(a^2-ax+x^2)-(a^2+ax+x^2)}{(a+x)-(a-x)}\right)\frac{(\sqrt{a+x}+\sqrt{a-x})}{(\sqrt{a^2-ax+x^2}+\sqrt{a^2+ax+x^2})}$

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

$=-\sqrt{a}$