Category: Limits

Question Number 182256 by mathlove last updated on 06/Dec/22 Answered by Ar Brandon last updated on 06/Dec/22 $$\mathscr{L}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\underset{{m}\rightarrow\infty} {\mathrm{lim}}\left(\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\underset{{k}=\mathrm{1}} {\overset{{mr}} {\sum}}\frac{{mn}^{\mathrm{2}} }{\left({m}^{\mathrm{2}}…

Question Number 116588 by bemath last updated on 05/Oct/20 $$\underset{x\to 0}{\lim }\frac{4\mathrm{x}}{2\mathrm{cosec}\mathrm{x}(1-\sqrt{\cos \mathrm{x}})}=?$$ Answered by bobhans last updated on 05/Oct/20 $$\underset{x\to 0}{\lim }\frac{4\mathrm{x}.\sin \mathrm{x}}{2(1-\sqrt{1-2\sin {}^2\left(\frac{\mathrm{x}}{2}\right)})}=$$$$\underset{{x}\rightarrow\mathrm{0}}…

Question Number 51005 by Tawa1 last updated on 23/Dec/18 Commented by maxmathsup by imad last updated on 23/Dec/18 $$letA\left(\theta \right)=\frac{sin(\theta -\frac{\pi }{6})}{\sqrt{3}-2cos\theta }changement\theta -\frac{\pi }{6}=xgive$$$$A\left(\theta \right)=\frac{sinx}{\sqrt{3}-2cos(x+\frac{\pi }{6})}=\frac{sinx}{\sqrt{3}-2(cosxcos\left(\frac{\pi }{6}\right)-sinxsin\left(\frac{\pi }{6}\right))}$$$$=\:\frac{{sinx}}{\:\sqrt{\mathrm{3}}−\mathrm{2}\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{cos}−\frac{\mathrm{1}}{\mathrm{2}}{sinx}\right)}\:=\frac{{sinx}}{\:\sqrt{\mathrm{3}}−\sqrt{\mathrm{3}}{cosx}\:+{sinx}}\:{but}\:{lim}_{\theta\rightarrow\frac{\pi}{\mathrm{6}}} \:\:={lim}_{{x}\rightarrow\mathrm{0}} \frac{{sinx}}{\:\sqrt{\mathrm{3}}−\sqrt{\mathrm{3}}{cosx}+{sinx}}…

Question Number 50954 by afachri last updated on 22/Dec/18 $$\mathrm{i}\mathrm{was}\mathrm{evaluating}$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{x(\sqrt[x]{x^{}}-1)}{\log x}$$$$\mathrm{and}\mathrm{got}0\mathrm{as}\mathrm{the}\mathrm{product}.\mathrm{is}\mathrm{it}\mathrm{true},\mathrm{My}\mathrm{Fellows}??$$ Commented by Abdo msup. last updated on…

Question Number 116417 by bemath last updated on 03/Oct/20 $$\underset{x\to 0}{\lim }\frac{\sqrt{1+\mathrm{x}\sin \mathrm{x}}-\sqrt{\cos 2\mathrm{x}}}{\tan {}^2\left(\frac{\mathrm{x}}{2}\right)}=?$$ Answered by Bird last updated on 03/Oct/20 $$f\left(x\right)=\frac{\sqrt{1+xsinx}-\sqrt{cos\left(2x\right)}}{{tan}^2\left(\frac{x}{2}\right)}$$$${we}\:{hsve}\:\sqrt{\mathrm{1}+{xsinx}}\sim\sqrt{\mathrm{1}+{x}^{\mathrm{2}}…