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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits4 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{n}\rightarrow\infty}\text{n}\sin\Big(\frac{\pi}{4\text{n}}\Big)\cos\Big(\frac{\pi}{4\text{n}}\Big)$
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Answer

$\lim\limits_{\text{n}\rightarrow\infty}\text{n}\sin\Big(\frac{\pi}{4\text{n}}\Big)\cos\Big(\frac{\pi}{4\text{n}}\Big)$
$=\lim\limits_{\text{n}\rightarrow\infty}2\Big(\text{n}\sin\frac{\pi}{4\text{n}}\cos\frac{\pi}{4\text{n}}\Big)\times\frac12$
$=\lim\limits_{\text{n}\rightarrow\infty}\text{n}\times\sin\frac{\pi}{2\text{n}}\times\frac12$
$\text{n}\rightarrow\infty,$ then $\frac{1}{\text{n}}\rightarrow0, $ let $\frac{1}{\text{n}}=\text{y}$
$=\frac12\lim\limits_{\frac{1}{\text{n}}\rightarrow\infty}\frac{1}{\text{y}}\sin\Big(\frac{\pi}{2}\Big)\Big(\frac{1}{\text{n}}\Big)$
$=\frac12\lim\limits_{{\text{y}}\rightarrow\infty}\frac{\sin\big(\frac\pi2\big)\text{y}}{\text{y}}$
$=\frac12\Bigg(\lim\limits_{{\text{y}}\rightarrow\infty}\frac{\sin\big(\frac{\pi\text{y}}2\big)}{\frac{\pi\text{y}}{2}}\Bigg)\times\frac\pi2$
$=\frac12\times1\times\frac\pi2$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1\Big]$
$=\frac\pi4$

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