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Limits — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSLimits5 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{x}^\circ-\sin2\text{x}^\circ}{\text{x}^3}$
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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{x}^\circ-\sin2\text{x}^\circ}{\text{x}^3}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{2\sin\frac{\pi\text{x}}{180}-\sin\frac{2\pi\text{x}}{180}}{\text{x}^3}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{2\sin\frac{\pi\text{x}}{180}-2\sin\frac{\pi\text{x}}{180}\cos\frac{\pi\text{x}}{180}}{\text{x}^3}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{2\sin\frac{\pi\text{x}}{180}\big(2\sin^2\frac{\pi\text{x}}{360}\big)}{\text{x}^3}$
$=4\bigg(\lim\limits_{\text{x} \rightarrow0}\frac{\sin\frac{\pi\text{x}}{180}}{\text{x}}\bigg)\times\bigg(\lim\limits_{\text{x} \rightarrow0}\frac{\sin\frac{\pi\text{x}}{360}}{\text{x}}\bigg)\times\bigg(\lim\limits_{\text{x} \rightarrow0}\frac{\sin\frac{\pi\text{x}}{360}}{\text{x}}\bigg)$
$=4\Bigg(\lim\limits_{\text{x} \rightarrow0}\frac{\sin\frac{\pi\text{x}}{180}}{\frac{\pi\text{x}}{180}}\times\frac{\pi}{180}\Bigg)\times\Bigg(\lim\limits_{\text{x} \rightarrow0}\frac{\sin\frac{\pi\text{x}}{360}}{\frac{\pi\text{x}}{360}}\times\frac{\pi}{360}\Bigg)\times\Bigg(\lim\limits_{\text{x} \rightarrow0}\frac{\sin\frac{\pi\text{x}}{360}}{\frac{\pi\text{x}}{360}}\times\frac{\pi}{360}\Bigg)$
$=4\times\frac{\pi}{180}\times\frac{\pi}{360}\times\frac{\pi}{360}$
$=\Big(\frac{\pi}{180}\Big)^3$

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