Category: Limits

Question Number 142277 by liberty last updated on 29/May/21 $$\underset{x\to 0}{\lim }\frac{1-e^{\sin x\ln \left(\cos x\right)}}{x^3}=?$$ Answered by Dwaipayan Shikari last updated on 29/May/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−{e}^{{sinx}\left(−\frac{{x}^{\mathrm{2}}…

Question Number 76630 by Rio Michael last updated on 28/Dec/19 $$\mathrm{two}\mathrm{sequences},\left(u_n\right)and\left(v_n\right),\mathrm{for}n\in \mathbb{N}\mathrm{is}\mathrm{defined}\mathrm{as}:$$$$\{\begin{array}{l}{u}_0=3\\ u_{n+1}=\frac{1}{2}(u_n+v_n)\end{array}\mathrm{and}\{\begin{array}{l}{v}_0=4\\ v_{n+1}=\frac{1}{2}(u_{n+1}+v_n)\end{array}$$$$\left.{a}\right)\:\mathrm{calculate}\:{u}_{\mathrm{1}}…

Question Number 142151 by iloveisrael last updated on 27/May/21 $$\underset{x\to \mathrm{\infty }}{\lim }\sin \sqrt{x+1}-\sin \sqrt{x}=?$$ Commented by gsk2684 last updated on 27/May/21 $$\underset{x\to \mathrm{\infty }}{\lim }\left(2\cos \frac{\sqrt{\mathrm{x}+1}+\sqrt{\mathrm{x}}}{2}\sin \frac{\sqrt{\mathrm{x}+1}-\sqrt{\mathrm{x}}}{2}\right)$$$$\mathrm{2}\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{cos}\:\frac{\sqrt{\mathrm{x}+\mathrm{1}}−\sqrt{\mathrm{x}}}{\mathrm{2}}\mathrm{sin}\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{x}+\mathrm{1}}+\sqrt{\mathrm{x}}}\right)…

Question Number 11036 by suci last updated on 08/Mar/17 $$l\underset{n\to \sim }{im}\underset{k=1}{\sum ^n}\frac{8n^2}{n^4+1}=\dots .?$$ Commented by FilupS last updated on 08/Mar/17 $${S}=\underset{{n}\rightarrow\infty}…

Question Number 76445 by john santu last updated on 27/Dec/19 $$howevaluate\underset{x\to 0}{\lim }\left(\frac{1-\sqrt{1+x^2}\times \cos x}{x^4}\right).$$ Commented by JDamian last updated on 27/Dec/19 $${What}\:{if}\:{you}\:{use}\:{L}'{Hopital}'{s}\:{rule}? \…