Category: Limits

Question Number 121527 by benjo_mathlover last updated on 09/Nov/20 $$\underset{x\to 0}{\lim }\frac{\sin \mathrm{x}\cos \mathrm{x}}{\sqrt{\pi +2\sin \mathrm{x}}-\sqrt{\pi }}?$$ Answered by Dwaipayan Shikari last updated on 09/Nov/20 $$\underset{x\to 0}{\lim }\frac{xcosx}{\pi +2sinx-\pi }(\sqrt{\pi +2sinx}+\sqrt{\pi })$$$$=\frac{{x}}{\mathrm{2}{sinx}}\left(\sqrt{\pi}+\sqrt{\pi}\right)\:\:\:\:\:\:\:\:\:\left({sinx}\rightarrow{x}\:\:\:{and}\:{x}\rightarrow\mathrm{0}\right)…

Question Number 121502 by bramlexs22 last updated on 08/Nov/20 $$\left(1\right)\underset{x\to 0}{\lim }\frac{\sqrt{1-\cos 2\mathrm{x}}}{\mathrm{x}\sqrt{2}}?$$$$\left(2\right)\int \cot {}^3\mathrm{x}\csc {}^3\mathrm{x}\mathrm{dx}$$$$$$ Commented by liberty last updated on…

Question Number 186983 by cortano12 last updated on 12/Feb/23 Answered by CElcedricjunior last updated on 12/Feb/23 $$\underset{x\to 0}{\lim }\frac{\mathit{x}\sqrt[3]{\mathit{x}-1}+\sqrt[4]{(\mathit{x}+1)}-1}{{\mathit{x}}^2\sqrt[3]{\mathit{x}+1}+\sqrt[4]{\mathit{x}+1}-1}=\mathit{k}$$$$\boldsymbol{{k}}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\boldsymbol{{x}}\left(−\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}}\boldsymbol{{x}}+\frac{\mathrm{1}}{\mathrm{9}}\boldsymbol{{x}}^{\mathrm{2}} \right)+\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{4}}\boldsymbol{{x}}−\frac{\mathrm{3}}{\mathrm{32}}\boldsymbol{{x}}^{\mathrm{2}} \right)−\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{2}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}\boldsymbol{{x}}−\frac{\:\:\mathrm{1}}{\mathrm{9}}\boldsymbol{{x}}^{\mathrm{2}}…

Question Number 55907 by gunawan last updated on 06/Mar/19 $$\mathrm{for}\mathrm{every}n\in \mathbb{N},f_n\left(x\right)=nx{(1-x^2)}^n,$$$$\mathrm{for}\mathrm{every}x,0⩽x⩽1$$$$\mathrm{and}{a}_n={\int }_0^1{f}_n\left(x\right)dx.$$$$\mathrm{If}\:\mathrm{S}_{\mathrm{n}} =\mathrm{sin}\:\left(\pi{a}_{{n}} \right),\:\mathrm{for}\:\mathrm{every}…

Question Number 55904 by gunawan last updated on 06/Mar/19 $$\mathrm{Let}{a}_i>0,{\mathrm{\forall }}_i=1,2,3,\dots 2016$$$$\mathrm{If}{(a_1{a}_2\dots a_{2016})}^{\frac{1}{2016}}=2$$$$\mathrm{then}$$$$\left(\mathrm{1}+{a}_{\mathrm{1}} \right)\left(\mathrm{1}+{a}_{\mathrm{2}} \right)\ldots\left(\mathrm{1}+{a}_{\mathrm{2016}} \right)\geqslant……