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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits2 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}\cos\text{x}}{3\text{x}}$
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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}\cos\text{x}}{3\text{x}}$ $=\frac13\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}\cos\text{x}}{{\text{x}}}{}$ $=\frac13\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{{\text{x}}}\times\lim\limits_{\text{x}\rightarrow0}\cos\text{x}$ $=\frac13\times1\times1$ $\Big[\because\ \lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1,\lim\limits_{\text{x}\rightarrow0}\cos0^\circ=1\Big]$ $=\frac13$

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