First slide

Introduction to limits

Question

The value of $lim_{x \to b} \frac{\sqrt{x - a} - \sqrt{b - a}}{x^{2} - b^{2}},$ for $b > a$

Moderate

A

$\frac{1}{4 b \sqrt{a - b}}$
B

$\frac{1}{4 b \sqrt{b - a}}$
C

$\frac{1}{4 a \sqrt{a - b}}$
D

$\frac{1}{b \sqrt{b - a}}$

Solution

We have,
$\begin{array}{l} lim_{x \to b} \frac{\sqrt{x - a} - \sqrt{b - a}}{x^{2} - b^{2}} \\ = lim_{x \to b} \frac{x - b}{(x - b) (x + b) {\sqrt{x - a} + \sqrt{b - a}}} \\ = lim_{x \to b} \frac{1}{(x + b) {\sqrt{x - a} + \sqrt{b - a}}} = \frac{1}{2 b {2 \sqrt{b - a}}} = \frac{1}{4 b \sqrt{b - a}} \end{array}$

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