First slide

Introduction to limits

Question

$lim_{x \to 1^{+}} \frac{\sqrt{x^{2} - 1} + \sqrt{x - 1}}{\sqrt{x^{2} - 1}}$ is equal to

Moderate

A

$\frac{1}{2}$
B

$\sqrt{2 + 1}$
C

1
D

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

Solution

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

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