Category: Limits

Question Number 194020 by mnjuly1970 last updated on 26/Jun/23 $$F_n=F_n{}_{-1}+F_{n-2}{F}_2=F_1=1$$$$F_n:1,1,2,3,5\dots$$$$\:\:\:\:\:\:\:{f}\left({x}\right)=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:{F}_{{n}} \:{x}^{\:{n}} \:=\:{x}\:+\:{x}^{\:\mathrm{2}}…

Question Number 193967 by Risandu last updated on 24/Jun/23 Answered by MM42 last updated on 24/Jun/23 $${lim}_{\theta \to \pi }\frac{4sin\theta cos\theta (cos\theta -sin3\theta )}{(\pi -\theta )(\pi +\theta )}$$$$let\theta =\pi -u$$$$\Rightarrow {lim}_{u\to 0}\frac{-4sinu\times cosu\times (sin3u-cosu)}{u\times (2\pi -u)}$$$$\:=\frac{\mathrm{2}}{\pi}\:\checkmark…

Question Number 193863 by horsebrand11 last updated on 21/Jun/23 $$\underset{x\to 0}{\lim }\frac{1-\frac{1}{2}{\mathrm{x}}^2-\cos \left(\frac{\mathrm{x}}{1-{\mathrm{x}}^2}\right)}{{\mathrm{x}}^4}=?$$ Answered by MM42 last updated on 21/Jun/23 $${u}\rightarrow\mathrm{0}\Rightarrow\mathrm{1}−{cosu}\:\sim\:\frac{\mathrm{1}}{\mathrm{2}}{u}^{\mathrm{2}} \…

Question Number 193803 by cortano12 last updated on 20/Jun/23 $$\underset{x\to 2}{\lim }\frac{\sqrt{11-\mathrm{x}}\cos \left(\frac{\pi }{\mathrm{x}-2}\right)}{\cot (\mathrm{x}-2)}=?$$ Answered by horsebrand11 last updated on 20/Jun/23 $$=\underset{x\to 2}{\lim }\frac{3\sin (\frac{\pi }{2}-\frac{\pi }{\mathrm{x}-2})}{\cot (\mathrm{x}-2)}$$$$\:=\:\mathrm{3}\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\mathrm{sin}\:\left(\mathrm{x}−\mathrm{2}\right)\:\mathrm{sin}\:\pi\left(\frac{\mathrm{x}−\mathrm{4}}{\mathrm{x}−\mathrm{2}}\right)…

Question Number 193532 by horsebrand11 last updated on 16/Jun/23 $$\underset{x\to 2}{\lim }\frac{1-\cos \pi \mathrm{x}}{{(2-\mathrm{x})}^2}=?$$ Answered by aba last updated on 16/Jun/23 $$\mathrm{let}\mathrm{t}=2-\mathrm{x}$$$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\pi\mathrm{x}}{\left(\mathrm{2}−\mathrm{x}\right)^{\mathrm{2}}…