Evaluate _ x 0 2x - 1 x - 1 - Vidyadip

Question

Evaluate $\mathop {\lim }\limits_{x \to 0} \frac{{\cos 2x - 1}}{{\cos x - 1}}$

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Answer

Here $\mathop {\lim }\limits_{x \to 0} \frac{{\cos 2x - 1}}{{\cos x - 1}}$$= \mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos 2x}}{{1 - \cos x}}$
$ = \mathop {\lim }\limits_{x \to 0} \frac{{2{{\sin }^2}x}}{{2{{\sin }^2}x/2}} = \mathop {\lim }\limits_{x \to 0} \frac{{{{(2\sin x/2\cos x/2)}^2}}}{{{{\sin }^2}x/2}}$
$ = \mathop {\lim }\limits_{x \to 0} \frac{{4{{\sin }^2}x/2{{\cos }^2}x/2}}{{{{\sin }^2}x/2}} = \mathop {\lim }\limits_{x \to 0} 4{\cos ^2}x/2$ = 4/2 = 2

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