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Question Number 28490 by A1B1C1D1 last updated on 26/Jan/18

$$\underset{\mathrm{x}\to \mathrm{a}}{\lim }\left(\frac{\cos \left(\mathrm{x}\right)-\cos \left(\mathrm{a}\right)}{\mathrm{x}-\mathrm{a}}\right)$$$$$$$${\mathrm{Don}}^{\prime }\mathrm{t}\mathrm{use}{\mathrm{L}}^{\prime }\mathrm{h}\overline{\mathrm{o}spital}\mathrm{rule}.$$

Answered by $@ty@m last updated on 27/Jan/18

$$Letx-a=t$$$$\Rightarrow asx\to a,t\to 0$$$$\therefore \underset{\mathrm{x}\to \mathrm{a}}{\lim }\left(\frac{\cos \left(\mathrm{x}\right)-\cos \left(\mathrm{a}\right)}{\mathrm{x}-\mathrm{a}}\right)$$$$=\underset{t\to 0}{\lim }\frac{\cos (t+a)-\cos a}{t}$$$$=\underset{t\to 0}{\lim }\frac{\cos t\cos a-\sin t\sin a-\cos a}{t}$$$$=\underset{t\to 0}{\lim }\frac{(\cos t-1)\cos a-\sin t\sin a}{t}$$$$=\underset{t\to 0}{\lim }\frac{(\cos t-1)\cos a}{t}-\underset{t\to 0}{\lim }\frac{\sin t\sin a}{t}$$$$=\underset{t\to 0}{\lim }\frac{-2\sin {}^2\frac{t}{2}\cos a}{t^2}\times t-\sin a\underset{t\to 0}{\lim }\frac{\sin t}{t}$$$$=0-\sin a$$$$=-\sin a$$

Commented by A1B1C1D1 last updated on 27/Jan/18

$$\mathrm{Thanks}$$

Answered by $@ty@m last updated on 27/Jan/18

$$Alternative:$$$$\underset{x\to a}{\lim }\frac{2\sin \frac{x+a}{2}\sin \frac{x-a}{2}}{x-a}$$$$\underset{x\to a}{\lim }\frac{\sin \frac{x+a}{2}\sin \frac{a-x}{2}}{\frac{x-a}{2}}$$$$\underset{x\to a}{\lim sin}\frac{x+a}{2}\times (-1\times \underset{x\to a}{\lim }\frac{\sin \frac{x-a}{2}}{\frac{x-a}{2}})$$$$\sin \frac{a+a}{2}\times (-1)\times 1$$$$=-\sin a$$

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