First slide

Introduction to limits

Question

The value of $lim_{x \to \infty} \sqrt{a^{2} x^{2} + a x + 1} - \sqrt{a^{2} x^{2} + 1}$ , is

Moderate

A

$\frac{1}{2}$
B

1
C

2
D

none of these

Solution

We have,
${\begin{array}{l} lim_{x \to \infty} \{\sqrt{a^{2} x^{2} + a x + 1} - \sqrt{a^{2} x^{2} + 1}\} \\ = lim_{x \to \infty} \frac{a x}{\sqrt{a^{2} x^{2} + a x + 1} + \sqrt{a^{2} x^{2} + 1}} \\ = lim_{x \to \infty} \frac{a}{\sqrt{a^{2} + \frac{a}{x} + \frac{1}{x^{2}}} + \sqrt{a^{2} + \frac{1}{x^{2}}}} = \frac{a}{2 a} = \frac{1}{2} \end{array}}^{}$

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