Evaluate the following limit: _ x 1 (1-x) ( 2 ) - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}(1-\text{x})\tan\Big(\frac{\pi\text{x}}{2}\Big)$

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Answer

$\lim\limits_{\text{x}\rightarrow1}(1-\text{x})\tan\Big(\frac{\pi\text{x}}{2}\Big)$
When x → 1, x - 1 → 0, let x - 1 = y, then y → 0
$=\lim\limits_{{(\text{x}-1)\rightarrow0}}-(\text{x}-1)\tan\frac{\pi\text{x}}{2}$
$=-\lim\limits_{\text{y}\rightarrow0}\text{y}\tan\frac{\text{y}}{2}(\text{y}+1)$
$=-\lim\limits_{\text{y}\rightarrow0}\text{y}\times\tan\Big(\frac{\pi}{2}+\frac{\pi}{2}\text{y}\Big)$
$=\lim\limits_{\text{y}\rightarrow0}\text{y}\times\cot\frac{\pi}{2}\text{y}$
$=\lim\limits_{\text{y}\rightarrow0}\frac{\text{y}}{\tan\frac{\pi\text{y}}{2}}$
$=\lim\limits_{\text{y}\rightarrow0}\frac{\frac{\pi\text{y}}{2}\times\frac{2}{\pi}}{\tan\frac{\pi\text{y}}{2}}$
$=\frac2\pi$

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