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Question Number 79065 by Khyati last updated on 22/Jan/20

$$].li\underset{x\to 0}{m}\frac{[sin(\alpha +\beta )x+sin(\alpha -\beta )x+sin2\alpha x}{cos2\beta x-cos2\alpha x}.x$$

Commented by jagoll last updated on 22/Jan/20

$$\underset{x\to 0}{\lim }\frac{[\sin (\alpha +\beta )\mathrm{x}+\sin (\alpha -\beta )\mathrm{x}+\sin 2\alpha \mathrm{x}]}{\cos 2\beta \mathrm{x}-\cos 2\alpha \mathrm{x}}\times \mathrm{x}?$$

Commented by mathmax by abdo last updated on 22/Jan/20

$$letconsider$$$$f\left(x\right)=\frac{sin(\alpha +\beta )x+sin(\alpha -\beta )x+sin\left(2\alpha x\right)}{cos\left(2\beta x\right)-cos\left(2\alpha x\right)}\times xforx\sim 0wehave$$$$f\left(x\right)\sim \frac{(\alpha +\beta )x+(\alpha -\beta )x+2\alpha x}{1-\frac{4{\beta }^2{x}^2}{2}-(1-\frac{4{\alpha }^2{x}^2}{2})}\times x\Rightarrow f\left(x\right)\sim \frac{x^2(\alpha +\beta +\alpha -\beta +2\alpha )}{x^2(2{\alpha }^2-2{\beta }^2)}$$$$\Rightarrow f\left(x\right)\sim \frac{4\alpha }{2{\alpha }^2-2{\beta }^2}\Rightarrow {lim}_{x\to 0}f\left(x\right)=\frac{2\alpha }{{\alpha }^2-{\beta }^2}(if\alpha \ne \beta )$$$$if\alpha =\beta fisnotdefined!$$

Commented by jagoll last updated on 23/Jan/20

$$\underset{x\to 0}{\lim }\frac{\mathrm{x}[2\sin \alpha \mathrm{xcos}\beta \mathrm{x}+\sin 2\alpha \mathrm{x}]}{-2\sin (\alpha +\beta )\mathrm{x}\sin (\beta -\alpha )\mathrm{x}}=$$$$\underset{x\to 0}{\lim }\frac{2\mathrm{xsin}\alpha \mathrm{x}+\mathrm{xsin}2\alpha \mathrm{x}}{-2\sin (\alpha +\beta )\mathrm{x}.\sin (\beta -\alpha )\mathrm{x}}=$$$$\frac{4\alpha }{(\alpha +\beta )(\alpha -\beta )}\times \frac{1}{2}=\frac{2\alpha }{{\alpha }^2-{\beta }^2}$$

Commented by Khyati last updated on 23/Jan/20

$$butansweris\frac{2\alpha }{{\alpha }^2-{\beta }^2}$$

Commented by jagoll last updated on 23/Jan/20

$$\mathrm{yes}.\mathrm{i}\mathrm{will}\mathrm{correct}\mathrm{my}\mathrm{typo}$$

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