Category: Limits

Question Number 29158 by abdo imad last updated on 04/Feb/18 $$find{lim}_{x\to 1}\frac{\sqrt{3+\sqrt{2x-1}}-2}{\sqrt{2+\sqrt{3x+1}}-\sqrt{x+3}}.$$ Commented by abdo imad last updated on 09/Feb/18 $$inthiscaseitsbettertousehospitaltheoremletput$$$${u}\left({x}\right)=\sqrt{\mathrm{3}+\sqrt{\mathrm{2}{x}−\mathrm{1}}}\:−\mathrm{2}\:{and}\:{v}\left({x}\right)=\sqrt{\mathrm{2}+\sqrt{\mathrm{3}{x}+\mathrm{1}}}\:\:−\sqrt{{x}+\mathrm{3}}…

Question Number 160218 by qaz last updated on 26/Nov/21 $$\underset{\mathrm{x}\to +\mathrm{\infty }}{\lim }\frac{{\int }_0^{\mathrm{x}}{\left(\arctan \mathrm{t}\right)}^2\mathrm{dt}}{\sqrt{1+{\mathrm{x}}^2}}=?$$ Answered by mathmax by abdo last updated on…

Question Number 160178 by cortano last updated on 25/Nov/21 $$\underset{x\to \pi }{\lim }\frac{\sin \frac{x}{2}-1}{\cos \left(\sin x\right)-1}=?$$ Commented by blackmamba last updated on 25/Nov/21 $$\underset{x\to \pi }{\lim }\frac{\sin \frac{x}{2}-1}{\cos \left(\sin x\right)-1}$$$$\:\:\:\:\:\:\:=\:\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\frac{{x}}{\mathrm{2}}−\mathrm{1}}{−\mathrm{2sin}\:^{\mathrm{2}}…

Question Number 160080 by qaz last updated on 24/Nov/21 $$\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{x}-{\int }_0^{\mathrm{x}}{\mathrm{e}}^{{\mathrm{t}}^2}\mathrm{dt}}{\mathrm{x}(1-\cos \mathrm{x})}=?$$ Answered by mr W last updated on 25/Nov/21…

Question Number 94528 by i jagooll last updated on 19/May/20 Commented by i jagooll last updated on 19/May/20 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}^{\mathrm{2}} \:\left\{\sqrt[{\mathrm{4}\:\:}]{\mathrm{1}−\frac{\mathrm{2}}{{x}}}−\mathrm{1}+\frac{\mathrm{2}}{{x}^{\mathrm{2}} }\right\}}{{x}^{\mathrm{2}} \left\{\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }−\sqrt[{\mathrm{4}\:\:}]{\mathrm{16}+\frac{\mathrm{7}}{{x}^{\mathrm{7}} }}\right\}}\:=\:\mathrm{0}…