Question Number 193532 by horsebrand11 last updated on 16/Jun/23
$$\:\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\pi\mathrm{x}}{\left(\mathrm{2}−\mathrm{x}\right)^{\mathrm{2}} }\:=? \\ $$
Answered by aba last updated on 16/Jun/23
$$\mathrm{let}\:\mathrm{t}=\mathrm{2}−\mathrm{x} \\ $$$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\pi\mathrm{x}}{\left(\mathrm{2}−\mathrm{x}\right)^{\mathrm{2}} }=\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\pi\left(\mathrm{2}+\mathrm{t}\right)}{\mathrm{t}^{\mathrm{2}} }=\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\pi\mathrm{t}}{\left(\pi\mathrm{t}\right)^{\mathrm{2}} }×\pi^{\mathrm{2}} =\frac{\pi^{\mathrm{2}} }{\mathrm{2}} \\ $$
Answered by cortano12 last updated on 16/Jun/23
$$\:\mathrm{by}\:\mathrm{L}'\mathrm{Hopital}\: \\ $$$$\:\mathrm{L}=\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\pi\mathrm{sin}\:\pi\mathrm{x}}{−\mathrm{2}\left(\mathrm{2}−\mathrm{x}\right)}\:=\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\pi^{\mathrm{2}} \mathrm{cos}\:\pi\mathrm{x}}{\mathrm{2}} \\ $$$$\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{2}} \\ $$