First slide

Introduction to limits

Question

The value of $lim_{x \to \infty} \frac{\sqrt{3 x^{2} - 1} + \sqrt{2 x^{2} - 1}}{4 x + 3}$ ,is

Moderate

A

$\frac{\sqrt{3} - \sqrt{2}}{4}$
B

$\frac{1}{4 (\sqrt{3} - \sqrt{2})}$
C

$\frac{\sqrt{3} - \sqrt{2}}{2}$
D

none of these

Solution

Dividing each term in the numerator and denominator by x, we get
$\begin{array}{l} lim_{x \to \infty} \frac{\sqrt{3 x^{2} - 1} + \sqrt{2 x^{2} - 1}}{4 x + 3} \\ = lim_{x \to \infty} \frac{\sqrt{3 - 1 / x^{2}} + \sqrt{2 - 1 / x^{2}}}{4 + 3 / x} = \frac{\sqrt{3} + \sqrt{2}}{4} = \frac{1}{4 (\sqrt{3} - \sqrt{2})} \end{array}$

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