Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}(\cos3\text{x}-\cos\text{x})}{\text{x}^3}$
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}(\cos3\text{x}-\cos\text{x})}{\text{x}^3}$
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Answer
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}(\cos3\text{x}-\cos\text{x})}{\text{x}^3}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}\Big(-2\sin\big(\frac{3\text{x}+\text{x}}{2}\big)\sin\big(\frac{3\text{x}-\text{x}}{2}\big)\Big)}{\text{x}^3}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}(-2\sin2\text{x}\sin\text{x})}{\text{x}^3}$
$=\frac{-2\lim\limits_{\text{x} \rightarrow0}\sin2\text{x}\times\lim\limits_{\text{x} \rightarrow0}\sin2\text{x}\times\lim\limits_{\text{x} \rightarrow0}\sin\text{x}}{\text{x}^3}$
$=-2\Big(\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}}{2\text{x}}\times2\Big)\times\Big(2\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}}{2\text{x}}\Big)\times\Big(\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}}{\text{x}}\Big)$
$=-2(1\times2)\times(2)\times(1)$
$=-8$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}\Big(-2\sin\big(\frac{3\text{x}+\text{x}}{2}\big)\sin\big(\frac{3\text{x}-\text{x}}{2}\big)\Big)}{\text{x}^3}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}(-2\sin2\text{x}\sin\text{x})}{\text{x}^3}$
$=\frac{-2\lim\limits_{\text{x} \rightarrow0}\sin2\text{x}\times\lim\limits_{\text{x} \rightarrow0}\sin2\text{x}\times\lim\limits_{\text{x} \rightarrow0}\sin\text{x}}{\text{x}^3}$
$=-2\Big(\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}}{2\text{x}}\times2\Big)\times\Big(2\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}}{2\text{x}}\Big)\times\Big(\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}}{\text{x}}\Big)$
$=-2(1\times2)\times(2)\times(1)$
$=-8$
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