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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits2 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{7\text{x}\cos\text{x}-3\sin\text{x}}{4\text{x}+\tan\text{x}}$
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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{7\text{x}\cos\text{x}-3\sin\text{x}}{4\text{x}+\tan\text{x}}$ $=\frac{\lim\limits_{\text{x}\rightarrow0}{7\cos\text{x}-\lim\limits_{\text{x}\rightarrow0}\frac{3\sin\text{x}}{\text{x}}}{}}{\lim\limits_{\text{x}\rightarrow0}4+\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}}$ $=\frac{7\times\lim\limits_{\text{x}\rightarrow0}\cos\text{x}-3\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}}{4+\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}}$ $=\frac{7\times1-3\times1}{4+1}$ $\Big[\because\ \lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1,\text{ also }\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}=1\Big]$ $=\frac{4}{5}$

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