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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits3 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{1-\tan\text{x}}{\text{x}-\frac{\pi}{4}}$
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Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{1-\tan\text{x}}{\text{x}-\frac{\pi}{4}}$ If $\text{x}\rightarrow\frac{\pi}{4},$ then $\text{x}-\frac\pi4\rightarrow0$ Let $\text{x}-\frac\pi4=\text{y}\Rightarrow\text{y}\rightarrow0$ $=\lim\limits_{\text{y}\rightarrow{0}}\frac{1-\tan\big(\text{y}+\frac{\pi}{4}\big)}{\text{y}}$ $=\lim\limits_{\text{y}\rightarrow{0}}\frac{1-\Bigg(\frac{\tan\text{y}+\tan\frac{\pi}{4}}{1-\tan\text{y}\tan\frac{\pi}{4}}\Bigg)}{\text{y}}$ $=\lim\limits_{\text{y}\rightarrow{0}}\frac{(1-\tan\text{y}-\tan\text{y}-1)}{\text{y}(1-\tan\text{y})}$ $\Big[\because\tan\frac{\pi}{4}=1\Big]$ $=\lim\limits_{\text{y}\rightarrow{0}}\frac{(-2\tan\text{y})}{(1-\tan\text{y})}$ $=-2\lim\limits_{\text{y}\rightarrow{0}}\frac{\tan\text{y}}{\text{y}}\times\frac{1}{\lim\limits_{\text{y}\rightarrow{0}}(1-\tan\text{y})}$ $=-2\times1\frac{1}{(1-0)}$ $\Big[\because\lim\limits_{\theta\rightarrow{0}}\frac{\sin\theta}{\theta}=1\Big]$ $=-2$

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