First slide

Introduction to limits

Question

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

Moderate

A

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

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

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

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

Solution

We have,
$\begin{array}{l} L = lim_{x \to 0} \frac{a - \sqrt{a^{2} - x^{2}} - \frac{x^{2}}{4}}{x^{4}} \\ \Rightarrow L = lim_{x \to 0} \frac{a - \sqrt{a^{2} - x^{2}} - \frac{x^{2}}{4}}{x^{4}} \\ \Rightarrow L = lim_{x \to 0} \frac{1}{x^{2}} \{\frac{1}{a + \sqrt{a^{2} - x^{2}}} - \frac{1}{4}\} \end{array}$
$\Rightarrow L = lim_{x \to 0} \frac{4 - a - \sqrt{a^{2} - x^{2}}}{4 x^{2} (a + \sqrt{a^{2} - x^{2})}}$
It is given that $L$ is finite.
$lim_{x \to 0} 4 - a - \sqrt{a^{2} - x^{2}} - 0 \Rightarrow 4 - a - a = 0 \Rightarrow a = 2$
Putting $a = 2$ , we obtain
$\begin{array}{l} L = lim_{x \to 0} \frac{2 - \sqrt{4 - x^{2}}}{4 x^{2} (2 + \sqrt{4 - x^{2}})} \\ \Rightarrow L = lim_{x \to 0} \frac{4 - (4 - x^{2})}{4 x^{2} {(2 + \sqrt{4 - x^{2}})}^{2}} \\ \Rightarrow L = lim_{x \to 0} \frac{1}{4 {(2 + \sqrt{4 - x^{2}})}^{2}} = \frac{1}{64} \end{array}$

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