Question

Find Lim x tends to a (cos x-cos a)/(x-a)

Answer 1

Answer :

Here, cosx - cosa

= 2 sin ($\frac{x+a}{2}$) sin ($\frac{a-x}{2}$)

Now,

$\lim(x \to\ a) \frac{ cosx - cosa}{x - a} \\ \\ = \lim(x \to \ a) \frac{2sin \frac{x + a}{2}sin \frac{a - x}{2} }{x - a} \\ \\ = - \lim(x \to a)sin \frac{x + a}{2} \times \lim(\frac{a - x}{2} \to\ 0) \frac{sin \frac{a - x}{2} }{ \frac{a - x}{2} } \\ \\ = - (sin \frac{a + a}{2}) \times 1 \\ \\ = - sin \frac{2a}{2} \\ \\ = - sina$

Here, $\lim(x \to\ 0) \frac{sinx}{x}=1$

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Answer 2

Given,

$\lim_{x \to a} \frac{ \cos x - \cos a}{x-a} \\ \\ = \lim_{x \to a} 2\frac{\sin\frac{(x+a)}{2}\sin\frac{(a-x)}{2}}{x-a} \\ \\= \lim_{x \to a} \sin\frac{(x+a)}{2} .\lim_{x \to a} \frac{sin\frac{(a-x)}{2}}{-\frac{(a-x)}{2}} \\ \\ = -\sin\frac{(a+a)}{2} *1 \\ \\ = -\sin a \ \ \ \textbf{Ans.}$