Category: Limits

Question Number 98955 by Ar Brandon last updated on 17/Jun/20 $$\underset{\mathrm{x}\to 0}{\lim }\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\cdot \cdot \cdot }}}}$$ Commented by mr W last updated on 17/Jun/20 $$=Q98858$$…

Question Number 98880 by Ar Brandon last updated on 16/Jun/20 $$\mathcal{G}\mathrm{iven}\mathrm{the}\mathrm{sequence}{\left({\mathrm{u}}_{\mathrm{n}}\right)}_{\mathrm{n}\in {\mathbb{N}}^{\ast }}\mathrm{defined}\mathrm{by}\{\begin{array}{l}\frac{1}{2^{\mathrm{n}}}\mathrm{if}\mathrm{n}\equiv 0\mathrm{mod}\left(3\right)\\ \frac{1}{3^{\mathrm{n}}}+1\mathrm{if}\mathrm{n}\equiv 1\mathrm{mod}\left(3\right)\\ \frac{{\mathrm{u}}_{\mathrm{n}-1}+{\mathrm{u}}_{\mathrm{n}+2}}{2}\mathrm{if}\mathrm{n}\equiv 2\mathrm{mod}\left(3\right)\end{array}$$$$\mathrm{a}\mathrm{\setminus }\mathcal{D}\mathrm{etermine}\mathrm{the}\mathrm{first}-8^{\mathrm{th}}\mathrm{terms}\mathrm{of}{\left({\mathrm{u}}_{\mathrm{n}}\right)}_{\mathrm{n}\in {\mathbb{N}}^{\ast }}$$$$\mathrm{b}\backslash\mathcal{S}\mathrm{how}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequences}\:\left(\mathrm{v}_{\mathrm{n}}…

Question Number 33313 by abdo imad last updated on 14/Apr/18 $$find{lim}_{x\to 0}\frac{ln(1+sinx)-sin\left(ln(1+x)\right)}{x^2}$$ Commented by abdo imad last updated on 17/Apr/18 $${let}\:{put}\:{u}\left({x}\right)={ln}\left(\mathrm{1}+{sinx}\right)\:−{sin}\left({ln}\left(\mathrm{1}+{x}\right)\right){and}\:{v}\left({x}\right)={x}^{\mathrm{2}} \…

Question Number 98833 by john santu last updated on 16/Jun/20 Commented by bobhans last updated on 16/Jun/20 $$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{3}^{\mathrm{n}} \:\left(\mathrm{1}+\left(\frac{\mathrm{1}}{\mathrm{6}^{\mathrm{n}} }\right)\right)^{\frac{\mathrm{1}}{\mathrm{n}}} \right)\:=\:\mathrm{3}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{6}^{\mathrm{n}} }\right)^{\frac{\mathrm{1}}{\mathrm{n}}} \…

Question Number 98823 by john santu last updated on 16/Jun/20 Commented by bramlex last updated on 16/Jun/20 $$L^{\prime }Hopital$$$$\underset{x\to 0}{\lim }\frac{\sin x+x\cos x}{3x^2}=\underset{x\to 0}{\lim }\frac{\cos x+\cos x-x\sin x}{6x}$$$$\underset{{x}\rightarrow\mathrm{0}}…