First slide

Introduction to limits

Question

$lim_{x \to - 1} \frac{\cos 2 - \cos 2 x}{x^{2} - | x |}$ is equal to

Moderate

A

2
B

sin 2
C

2 sin2
D

none of these

Solution

We have,
$\begin{array}{l} lim_{x \to - 1} \frac{\cos 2 - \cos 2 x}{x^{2} - | x |} \\ = lim_{x \to - 1} \frac{2 \sin (1 + x) \sin (x - 1)}{x^{2} + x} [∵ | x | = - x for x < 0] \\ = 2 lim_{x \to - 1} \frac{\sin (x + 1)}{x + 1} \times \frac{\sin (x - 1)}{x} \\ = 2 \times 1 \times \frac{\sin (- 2)}{(- 1)} = 2 \sin 2 . \end{array}$

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