Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos3\text{x}-\cos7\text{x}}{\text{x}^2}$
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos3\text{x}-\cos7\text{x}}{\text{x}^2}$
✓
Answer
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos3\text{x}-\cos7\text{x}}{\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\Bigg(\frac{-2\sin\big(\frac{3\text{x}+7\text{x}}{2}\big)\sin\big(\frac{3\text{x}-7\text{x}}{2}\big)}{\text{x}^2}\Bigg)$
$=\lim\limits_{\text{x}\rightarrow0}\Bigg(\frac{-2\sin5\text{x}\sin\big(\frac{-4\text{x}}{2}\big)}{\text{x}^2}\Bigg)$
$=\Big(\lim\limits_{\text{x}\rightarrow0}\frac{-2\sin5\text{x}}{\text{x}}\Big)\times\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\sin(-2\text{x})}{\text{x}}\Big)$
$=\Big(-2\lim\limits_{\text{x}\rightarrow0}\frac{\sin5\text{x}}{5\text{x}}\Big)\times\Big(-1\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}}{2\text{x}}\times2\Big)$
$=(-2\times5)(-1\times2)$
$=20$
$=\lim\limits_{\text{x}\rightarrow0}\Bigg(\frac{-2\sin\big(\frac{3\text{x}+7\text{x}}{2}\big)\sin\big(\frac{3\text{x}-7\text{x}}{2}\big)}{\text{x}^2}\Bigg)$
$=\lim\limits_{\text{x}\rightarrow0}\Bigg(\frac{-2\sin5\text{x}\sin\big(\frac{-4\text{x}}{2}\big)}{\text{x}^2}\Bigg)$
$=\Big(\lim\limits_{\text{x}\rightarrow0}\frac{-2\sin5\text{x}}{\text{x}}\Big)\times\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\sin(-2\text{x})}{\text{x}}\Big)$
$=\Big(-2\lim\limits_{\text{x}\rightarrow0}\frac{\sin5\text{x}}{5\text{x}}\Big)\times\Big(-1\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}}{2\text{x}}\times2\Big)$
$=(-2\times5)(-1\times2)$
$=20$
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