Question Number 186758 by EnterUsername last updated on 09/Feb/23 $$\mathrm{If}\:\left[{t}\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{part}\:\mathrm{of}\:{t},\:\mathrm{then}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left[{x}\:\mathrm{sin}\:\pi{x}\right] \\ $$$$\left(\mathrm{A}\right)\:\:\mathrm{equals}\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\:\mathrm{equals}\:−\mathrm{1} \\ $$$$\left(\mathrm{C}\right)\:\:\mathrm{equals}\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$ Answered by Gazella thomsonii last updated on 10/Feb/23…
Question Number 186748 by EnterUsername last updated on 09/Feb/23 $$\mathrm{Let}\:{f}:\mathbb{R}^{+} \rightarrow\mathbb{R}^{+} \:\mathrm{be}\:\mathrm{a}\:\mathrm{function}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{relation} \\ $$$${f}\left({x}.{f}\left(\mathrm{y}\right)\right)={f}\left({x}\mathrm{y}\right)+{x}\:\mathrm{for}\:\mathrm{all}\:{x},\:\mathrm{y}\:\in\mathbb{R}^{+} .\:\mathrm{Then} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\left({f}\left({x}\right)\right)^{\mathrm{1}/\mathrm{3}} −\mathrm{1}}{\left({f}\left({x}\right)\right)^{\mathrm{1}/\mathrm{2}} −\mathrm{1}}\right)= \\ $$$$\left(\mathrm{A}\right)\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left(\mathrm{C}\right)\:\:\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\:\frac{\mathrm{3}}{\mathrm{2}} \\…
Question Number 55642 by gunawan last updated on 01/Mar/19 $$\mathrm{Prove}\:\mathrm{the}\:\mathrm{following}\:\mathrm{statements}: \\ $$$$\mathrm{If}\:\mathrm{for}\:\mathrm{every}\:{n}\:,\:{f}_{{n}} \:\mathrm{form}\:\mathrm{ascend}\:\mathrm{function} \\ $$$$\mathrm{and}\:\left\{{f}_{{n}} \right\}\:\mathrm{uniform}\:\mathrm{convergences} \\ $$$$\mathrm{to}\:{f}\:\mathrm{at}\:\left[{a},\:{b}\right],\:\mathrm{then} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\int_{{a}} ^{{b}} {f}_{{n}} \left({x}\right)\:{dx}\:\rightarrow\int_{{a}} ^{{b}}…
Question Number 55643 by gunawan last updated on 01/Mar/19 $$\mathrm{known}\:\mathrm{function}\:{f} \\ $$$$\mathrm{diferensiable}\:\mathrm{continues}\:\mathrm{at}\:\left[{a},\:{b}\right] \\ $$$$\mathrm{If}\:{f}\left({a}\right)={f}\left({b}\right)=\mathrm{0} \\ $$$$\mathrm{and}\: \\ $$$$\int_{{a}} ^{{b}} \left[{f}\left({x}\right)\right]^{\mathrm{2}} {dx}=\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\int_{{a}}…
Question Number 55640 by gunawan last updated on 01/Mar/19 $$\mathrm{known}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{sequence} \\ $$$$\left\{{a}_{{n}} \right\}\:\mathrm{and}\:\left\{{b}_{{n}} \right\}\:\mathrm{both}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{convergences}\:\mathrm{to}\:\mathrm{0}. \\ $$$$\mathrm{If}\:\left\{{b}_{{n}} \right\}\:\mathrm{monotonous}\:\mathrm{descend} \\ $$$$\mathrm{and}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{a}_{{n}+\mathrm{1}} −{a}_{{n}} }{{b}_{{n}+\mathrm{1}} −{b}_{{n}}…
Question Number 55641 by gunawan last updated on 01/Mar/19 $$\mathrm{Studies}\:\mathrm{of}\:\mathrm{convergences} \\ $$$$\mathrm{the}\:\mathrm{numbers}\:\mathrm{real}\:\mathrm{sequence}\:\left\{{x}_{{n}} \right\}, \\ $$$$\mathrm{with}\:{x}_{\mathrm{1}} =\mathrm{1}\:\mathrm{and}\:{x}_{{n}+\mathrm{1}} =\frac{{x}_{{n}} ^{\mathrm{2}} +\mathrm{2}}{\mathrm{2}{x}_{{n}} },\:{n}\geqslant\mathrm{1} \\ $$$$ \\ $$$$ \\…
Question Number 55639 by gunawan last updated on 01/Mar/19 $$\mathrm{Known}\:{a}\:\in\:\mathbb{R}\:\mathrm{and} \\ $$$$\mathrm{function}\:{f}\::\:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{satiesfied} \\ $$$$\mid{xf}\left({x}\right)+{a}\mid\:<\:\mathrm{sin}^{\mathrm{2}} \:\left({x}−{a}\right).\: \\ $$$$\mathrm{For}\:\mathrm{all}\:{x}\:\in\:\mathbb{R} \\ $$$$\mathrm{value}\:\mathrm{of}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{f}\left({x}\right)\:.. \\ $$ Answered by kaivan.ahmadi…
Question Number 55637 by gunawan last updated on 01/Mar/19 $$\mathrm{Value}\:\mathrm{of}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{n}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{2}{x}^{{n}} }{{x}+{x}^{\mathrm{2}{n}+\mathrm{1}} }\:{dx}=.. \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 01/Mar/19 $$\int_{\mathrm{0}}…
Question Number 121172 by benjo_mathlover last updated on 05/Nov/20 $$\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}−\mathrm{x}+\mathrm{1}}{\mathrm{x}+\mathrm{1}}\:? \\ $$$$ \\ $$ Answered by TANMAY PANACEA last updated on 05/Nov/20 $$\underset{{x}\rightarrow\infty}…
Question Number 55635 by gunawan last updated on 01/Mar/19 $$\mathrm{Series}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\Sigma}}\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }=.. \\ $$ Commented by Joel578 last updated on 01/Mar/19 $${Ans}\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\…