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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits2 Marks
Question
Evaluate the following limit: $\lim\limits_{\theta\rightarrow0}\frac{1-\cos4\theta}{1-\cos6\theta}$
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Answer

$\lim\limits_{\theta\rightarrow0}\frac{1-\cos4\theta}{1-\cos6\theta}$ $=\lim\limits_{\theta \rightarrow0}\frac{2\sin^22\theta}{2\sin^23\theta}$ $=\lim\limits_{\theta \rightarrow0}\frac{(\sin2\theta)^2}{(\sin3\theta)^2}$ $=\frac{\lim\limits_{\theta \rightarrow0}\big(\frac{\sin2\theta}{2\theta}\big)^2\times4\theta^2}{\lim\limits_{\theta \rightarrow0}\big(\frac{\sin3\theta}{3\theta}\big)^2\times9\theta^2}$ $=\frac{1\times4\theta^2}{1\times9\theta^2}$ $=\frac{4}{9}$

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