Category: Limits

Question Number 149184 by liberty last updated on 03/Aug/21 $$\mathrm{\Omega }=\underset{x\to 1}{\lim }\frac{\tan \sqrt{3\mathrm{x}+2}-\tan \sqrt{2\mathrm{x}+3}}{\tan \sqrt{5\mathrm{x}+4}-\tan \sqrt{4\mathrm{x}+5}}=?$$ Answered by EDWIN88 last updated on 03/Aug/21 $$\Omega=\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\frac{\mathrm{3}}{\mathrm{2}\sqrt{\mathrm{3}{x}+\mathrm{2}}}\:\mathrm{sec}\:^{\mathrm{2}} \:\sqrt{\mathrm{3}{x}+\mathrm{2}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}{x}+\mathrm{3}}}\mathrm{sec}\:^{\mathrm{2}} \sqrt{\mathrm{2}{x}+\mathrm{3}}}{\frac{\mathrm{5}}{\mathrm{2}\sqrt{\mathrm{5}{x}+\mathrm{4}}}\mathrm{sec}\:^{\mathrm{2}} \sqrt{\mathrm{5}{x}+\mathrm{4}}−\frac{\mathrm{2}}{\:\sqrt{\mathrm{4}{x}+\mathrm{5}}}\mathrm{sec}\:^{\mathrm{2}}…

Question Number 83492 by jagoll last updated on 03/Mar/20 $$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{2\sqrt{\mathrm{x}}}$$$$\underset{\mathrm{t}\to 0}{\lim }\frac{\mathrm{f}(2\mathrm{x}-\mathrm{t})+\mathrm{f}(2\mathrm{x}-2\mathrm{t})-2\mathrm{f}(2\mathrm{x}+\mathrm{t})}{\mathrm{t}}$$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Question Number 17957 by Arnab Maiti last updated on 13/Jul/17 $$\mathrm{prove}\mathrm{that}{(1+\frac{\mathrm{a}}{\mathrm{x}})}^{\mathrm{n}}=(1+\frac{\mathrm{an}}{\mathrm{x}}),\mathrm{x}\gg \mathrm{a}$$ Answered by 42 last updated on 13/Jul/17 $$\mathrm{Expand}\mathrm{using}\mathrm{binomial}\mathrm{theorem}:$$$$\mathrm{1}\:+\:{n}\left(\frac{{a}}{{x}}\right)\:+\:\frac{{n}\left({n}\:−\:\mathrm{1}\right)}{\mathrm{2}}\left(\frac{{a}}{{x}}\right)^{\mathrm{2}} \:+\:……

Question Number 148992 by mathlove last updated on 02/Aug/21 Answered by gsk2684 last updated on 02/Aug/21 $$\approx \underset{x\to 0}{\lim }\frac{(\sin x+\frac{\sin {}^{2}x}{2!}+..)-(\sin 3x+\frac{\sin {}^{2}3x}{2!}+..)}{2x}$$$$=\frac{1}{2}\{(1+0+\dots )-(3+0+..)\}$$$$=−\mathrm{1} \…

Question Number 83401 by jagoll last updated on 02/Mar/20 $$\underset{x\to \mathrm{\infty }}{\lim }(6-8\mathrm{xsin}\left(\frac{3}{\mathrm{x}}\right))=?$$ Commented by mathmax by abdo last updated on 02/Mar/20 $$\mathrm{6}−\mathrm{8}{x}\:{sin}\left(\frac{\mathrm{3}}{{x}}\right)\sim\mathrm{6}−\mathrm{8}{x}×\frac{\mathrm{3}}{{x}}\:\Rightarrow{lim}_{{x}\rightarrow\infty} \left(\mathrm{6}−\mathrm{8}{x}\:{sin}\left(\frac{\mathrm{3}}{{x}}\right)\right)=\mathrm{6}−\mathrm{24}=−\mathrm{18} \…