Evaluate the following limit: _ x 0 ( + )x+ ( - )x+ 2 ^2 - ^2

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\{\sin(\alpha+\beta)\text{x}+\sin(\alpha-\beta)\text{x}+\sin2\alpha\text{x}\}}{\cos^2\beta\text{x}-\cos^2\alpha\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\{\sin(\alpha+\beta)\text{x}+\sin(\alpha-\beta)\text{x}+\sin2\alpha\text{x}\}}{\cos^2\beta\text{x}-\cos^2\alpha\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\Big\{2\sin\frac{(\alpha+\beta+\alpha-\beta)}{2}\times\cos\frac{(\alpha+\beta-\alpha+\beta)}{2}\times+2\sin\alpha\cos\alpha\text{x}\Big\}}{(\cos\beta\text{x}-\cos\alpha\text{x})(\cos\beta\text{x}+\cos\alpha\text{x})}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big\{2\sin\alpha\text{x}\cos\beta\text{x}+2\sin\alpha\text{x}\cos\alpha\text{x}\big\}}{(\cos\beta\text{x}-\cos\alpha\text{x})(\cos\beta\text{x}+\cos\alpha\text{x})}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\alpha\text{x}(\cos\beta\text{x}+\cos\alpha\text{x})}{(\cos\beta\text{x}-\cos\alpha\text{x})(\cos\beta\text{x}+\cos\alpha\text{x})}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\alpha\text{x}}{(\cos\beta\text{x}-\cos\alpha\text{x})}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\alpha\text{x}}{\Big(1-2\sin^2\big(\frac{\beta\text{x}}{2}\big)-1+2\sin^2\big(\frac{\alpha\text{x}}{2}\big)\Big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\alpha\text{x}}{2\sin^2\big(\frac{\alpha\text{x}}{2}\big)-2\sin^2\big(\frac{\beta\text{x}}{2}\big)}$
$=\frac{2\alpha}{\alpha^2-\beta^2}$