Question

Let $L=\underset{x\to 0}{\lim }\frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4},a>0.$ 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, $L=\underset{x\to 0}{\lim }\frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4}$ [form $\frac{0}{0}t]$
$=\underset{x\to 0}{\lim }\frac{0-\frac{\left(0-2x\right)}{2\sqrt{a^2-x^2}}-\frac{2x}{4}}{4x^3}$
$=\underset{x\to 0}{\lim }\frac{x\left(\frac{1}{\sqrt{a^2-x^2}}-\frac{1}{2}\right)}{4x^3}$
$=\underset{x\to 0}{\lim }\frac{\frac{1}{\sqrt{a^2-x^2}}-\frac{1}{2}}{4x^2}$
For limit to be exist, $a=2$