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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits3 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{\text{f(x)}-\text{f}\big(\frac\pi4\big)}{\text{x}-\frac\pi4},$ where $\text{f(x)}=\sin2\text{x}$
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Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\Bigg(\frac{\sin2\text{x}-\sin2\big(\frac\pi4\big)}{\text{x}-\frac\pi4}\Bigg)$ $\big[\because$ given $\text{f(x)}=\sin2\text{x}\big]$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\bigg(\frac{\sin2\text{x}-\sin\frac{\pi}{2}}{\text{x}-\frac\pi4}\bigg)$
$\Rightarrow\text{x}\rightarrow\frac{\pi}{4}\Rightarrow\text{x}-\frac{\pi}{4}\rightarrow0,$ let $\text{x}-\frac\pi4=\text{y}$
$=\lim\limits_{\text{y}\rightarrow{0}}\Bigg(\frac{\sin2\big(\text{y}+\frac{\pi}{4}\big)-1}{\text{y}}\Bigg)$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\sin\big(\frac\pi2+2\text{y}\big)-1}{\text{y}}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\cos2\text{y}-1}{\text{y}}$
$=-\lim\limits_{\text{y}\rightarrow{0}}\frac{1-\cos2\text{y}}{\text{y}}$
$=-\lim\limits_{\text{y}\rightarrow{0}}\frac{2\sin^2\text{y}}{\text{y}}$
$=-2\Big(\lim\limits_{\text{y}\rightarrow{0}}\frac{\sin\text{y}}{\text{y}}\Big)^2\times\text{y}$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1\Big]$
$=-2\times0$
$=0$

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