Question

Let $ L = \displaystyle\lim_{x\rightarrow 0} \frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4}, a > 0. $ If $L$ is finite, then

• A. $ a = 2 $
• B. $ a = 1 $
• C. $ a = \frac{1}{3} $
• D. None of these

Answer 1

Answer: (A)

Given, $L=\displaystyle\lim_{x\to0} \frac{a-\sqrt{a^{2}-x^{2}}-\frac{x^{2}}{4}}{x^{4}}$ [form $\frac{0}{0}t]$
$=\displaystyle\lim_{x\to0} \frac{0-\frac{\left(0-2x\right)}{2\sqrt{a^{2}-x^{2}}}-\frac{2x}{4}}{4x^{3}}$
$=\displaystyle\lim_{x\to0} \frac{x\left(\frac{1}{\sqrt{a^{2}-x^{2}}}-\frac{1}{2}\right)}{4x^{3}}$
$=\displaystyle\lim_{x\to0} \frac{\frac{1}{\sqrt{a^{2}-x^{2}}}-\frac{1}{2}}{4x^{2}}$
For limit to be exist, $a = 2$