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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits3 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\tan\text{x}}{1-\cos\text{x}}$
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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\tan\text{x}}{1-\cos\text{x}}$ $=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}\sin\text{x}}{\cos\text{x}(1-\cos\text{x})}$ $=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}\sin\text{x}}{\cos\text{x}\big(2\sin^2\frac{\text{x}}{2}\big)}$ $=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}\big(2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}\big)}{\cos\text{x}\big(2\sin^2\frac{\text{x}}{2}\big)}$ $=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}\cos\frac{\text{x}}{2}}{\cos\text{x}\frac{\sin{\text{x}}{}}{2}}$ $=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}\cos\frac{\text{x}}{2}}{\cos\text{x}\frac{\sin\text{x}}{2}}$ $=\lim\limits_{\text{x} \rightarrow0}\frac{1}{\cos\text{x}\times\frac{\frac{\tan\text{x}}{2}}{\text{x}}}$ $=\lim\limits_{\text{x} \rightarrow0}\frac{1}{\cos\text{x}}\times\frac{1}{\cos\text{x}\times\frac{\frac{\tan\text{x}}{2}}{\frac{\text{x}}{2}}\times\frac12}$ $=1\times1\times2$ $=2$

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