Let L=limx→0 a−a2−x2−x2/4×4,a>0 If L is finite, then

# Let $L=\underset{x\to 0}{lim} \frac{a-\sqrt{{a}^{2}-{x}^{2}}-{x}^{2}/4}{{x}^{4}},a>0$ If $L$ is finite, then

1. A

$a=2,L=\frac{1}{64}$

2. B

$a=1,L=\frac{1}{64}$

3. C

$a=3,L=\frac{1}{32}$

4. D

$a=1,L=\frac{1}{32}$

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### Solution:

We have,

It is given that $L$ is finite.

$\underset{x\to 0}{lim} 4-a-\sqrt{{a}^{2}-{x}^{2}}-0⇒4-a-a=0⇒a=2$

Putting $a=2$, we obtain

$\begin{array}{l}L=\underset{x\to 0}{lim} \frac{2-\sqrt{4-{x}^{2}}}{4{x}^{2}\left(2+\sqrt{4-{x}^{2}}\right)}\\ ⇒L=\underset{x\to 0}{lim} \frac{4-\left(4-{x}^{2}\right)}{4{x}^{2}{\left(2+\sqrt{4-{x}^{2}}\right)}^{2}}\\ ⇒L=\underset{x\to 0}{lim} \frac{1}{4{\left(2+\sqrt{4-{x}^{2}}\right)}^{2}}=\frac{1}{64}\end{array}$

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