First slide

Introduction to limits

Question

$lim_{x \to 1} \frac{\sqrt{1 - \cos 2 (x - 1)}}{x - 1}$

Moderate

A

exists and it equals $\sqrt{2}$
B

exists and it equals $- \sqrt{2}$
C

does not exist because $(x - 1) \to 0$
D

does not exist because left hand limit is not equal to right hand limit

Solution

We have,
$\begin{array}{l} lim_{x \to 1} \frac{\sqrt{1 - \cos 2 (x - 1)}}{x - 1} \\ = lim_{x \to 1} \frac{\sqrt{2 \sin^{2} (x - 1)}}{x - 1} = lim_{x \to 1} \sqrt{2} \frac{| \sin (x - 1) |}{x - 1} \\ ∴ (LHL at x = 1) = \sqrt{2} lim_{x \to 1^{-}} \frac{| \sin (x - 1) |}{x - 1} \\ = \sqrt{2} lim_{x \to 1} - \frac{\sin (x - 1)}{x - 1} = - \sqrt{2} \end{array}$
and,
$\begin{array}{l} (RHL at x = 1) = \sqrt{2} lim_{x \to 1^{+}} \frac{\sin (x - 1) ∣}{x - 1} \\ = \sqrt{2} lim_{x \to 1} \frac{\sin (x - 1)}{x - 1} = \sqrt{2} \end{array}$
Clearly, $(LHL at x = 1) \neq (RHL at x = 1)$
So, $lim_{x \to 1} \frac{\sqrt{1 - \cos 2 (x - 1)}}{x - 1}$ does not exist.

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