Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits4 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow1}(1-\text{x})\tan\Big(\frac{\pi\text{x}}{2}\Big)$
✓
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 $=\lim\limits_{{(\text{x}-1)\rightarrow0}}-(\text{x}-1)\tan\frac{\pi\text{x}}{2}$ $=-\lim\limits_{\text{y}\rightarrow0}\text{y}\tan\frac{\text{y}}{2}(\text{y}+1)$ $=-\lim\limits_{\text{y}\rightarrow0}\text{y}\times\tan\Big(\frac{\pi}{2}+\frac{\pi}{2}\text{y}\Big)$ $=\lim\limits_{\text{y}\rightarrow0}\text{y}\times\cot\frac{\pi}{2}\text{y}$ $=\lim\limits_{\text{y}\rightarrow0}\frac{\text{y}}{\tan\frac{\pi\text{y}}{2}}$ $=\lim\limits_{\text{y}\rightarrow0}\frac{\frac{\pi\text{y}}{2}\times\frac{2}{\pi}}{\tan\frac{\pi\text{y}}{2}}$ $=\frac2\pi$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.