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Question Number 84460 by jagoll last updated on 13/Mar/20

$$\underset{x\to 0}{\lim }\frac{\sin (2+\mathrm{x})-\sin (2-\mathrm{x})}{\mathrm{x}}$$

Commented by jagoll last updated on 13/Mar/20

$$\underset{x\to 0}{\lim }\frac{\cos (2+\mathrm{x})+\cos (2-\mathrm{x})}{1}$$$$=2\cos 2$$

Commented by mathmax by abdo last updated on 13/Mar/20

$$letf\left(x\right)=\frac{sin(2+x)-sin(2-x)}{x}\Rightarrow$$$$f\left(x\right)=\frac{sin2cosx+cos2sinx-(sin2cosx-cos2sinx)}{x}$$$$=\frac{2cos2sinx}{x}\sim 2cos2(x\to 0)\Rightarrow {lim}_{x\to 0}f\left(x\right)=2cos2$$

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