Question Number 184880 by mathlove last updated on 13/Jan/23
$$\underset{{x}\rightarrow\frac{\mathrm{1}}{\pi}} {\mathrm{lim}}\:\:\frac{\pi^{−\mathrm{1}} −{x}}{\frac{\mathrm{1}−\left({x}\pi\right)^{\mathrm{2}} }{\pi^{\mathrm{2}} }}=? \\ $$
Answered by aba last updated on 13/Jan/23
$$\underset{{x}\rightarrow\frac{\mathrm{1}}{\pi}} {\mathrm{lim}}\frac{\pi^{−\mathrm{1}} −\mathrm{x}}{\frac{\mathrm{1}−\left(\pi\mathrm{x}\right)^{\mathrm{2}} }{\pi^{\mathrm{2}} }}=\underset{{x}\rightarrow\frac{\mathrm{1}}{\pi}} {\mathrm{lim}}\frac{\pi^{\mathrm{2}} \left(\pi^{−\mathrm{1}} −\mathrm{x}\right)}{\mathrm{1}−\left(\pi\mathrm{x}\right)^{\mathrm{2}} }=\underset{{x}\rightarrow\frac{\mathrm{1}}{\pi}} {\mathrm{lim}}\frac{\pi\left(\mathrm{1}−\pi\mathrm{x}\right)}{\mathrm{1}−\left(\pi\mathrm{x}\right)^{\mathrm{2}} }=\underset{{x}\rightarrow\frac{\mathrm{1}}{\pi}} {\mathrm{lim}}\frac{\pi}{\mathrm{1}+\pi\mathrm{x}}=\frac{\pi}{\mathrm{2}} \\ $$