First slide

Introduction to limits

Question

The value of $lim_{x \to a} \frac{\sqrt{3 x - a} - \sqrt{x + a}}{x - a}$ is

Easy

A

$- \frac{1}{\sqrt{2} a}$
B

$\frac{1}{\sqrt{2} a}$
C

$\frac{1}{\sqrt{3} a}$
D

$\frac{1}{\sqrt{2}}$

Solution

$\begin{array}{l} lim_{x \to a} \frac{\sqrt{3 x - a} - \sqrt{x + a}}{x - a} \\ = lim_{x \to a} \frac{\sqrt{3 x - a} - \sqrt{x + a}}{x - a} \times \frac{\sqrt{3 x - a} + \sqrt{x + a}}{\sqrt{3 x - a} + \sqrt{x + a}} \\ = lim_{x \to a} \frac{(3 x - a) - (x + a)}{(x - a) (\sqrt{3 x - a} + \sqrt{x + a})} \\ = lim_{x \to a} \frac{2 (x - a)}{(x - a) (\sqrt{3 x - a} + \sqrt{x + a})} \\ = \frac{2}{\sqrt{3 a - a} + \sqrt{a + a}} \\ = \frac{2}{2 \sqrt{2} a} = \frac{1}{\sqrt{2} a} \end{array}$

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