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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits3 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sin(2+\text{x})-\sin(2-\text{x})}{\text{x}}$
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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin(2+\text{x})-\sin(2-\text{x})}{\text{x}}$ $=\lim\limits_{\text{x}\rightarrow0}2\cos\frac{\Big(\frac{{2+\text{x}+2-\text{x}}{2}}{\text{x}}\Big)\times\sin\Big(\frac{2+\text{x}-2+\text{x}}{2}\Big)}{{\text{x}}}$ $=2\lim\limits_{\text{x}\rightarrow0}\frac{\cos(2)\times\sin\text{x}}{\text{x}}$ $=2\lim\limits_{\text{x}\rightarrow0}\cos2\times\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}$ $\Big[\because\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1\Big]$ $=2\cos2\times1$ $=2\cos2$

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