Question Number 55907 by gunawan last updated on 06/Mar/19 $$\mathrm{for}\:\mathrm{every}\:{n}\:\in\:\mathbb{N}\:,\:{f}_{{n}} \left({x}\right)={nx}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} , \\ $$$$\mathrm{for}\:\mathrm{every}\:{x},\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1} \\ $$$$\mathrm{and}\:{a}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} {f}_{{n}} \left({x}\right)\:{dx}. \\ $$$$\mathrm{If}\:\mathrm{S}_{\mathrm{n}} =\mathrm{sin}\:\left(\pi{a}_{{n}} \right),\:\mathrm{for}\:\mathrm{every}…
Question Number 55906 by gunawan last updated on 06/Mar/19 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\overset{{n}} {\underset{{k}=\mathrm{1}} {\sum}}\frac{\mathrm{8}{n}^{\mathrm{2}} }{{n}^{\mathrm{4}} +\mathrm{1}}=.. \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 06/Mar/19 $${T}_{{n}}…
Question Number 55904 by gunawan last updated on 06/Mar/19 $$\mathrm{Let}\:{a}_{{i}} >\mathrm{0},\:\forall_{{i}} =\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\ldots\mathrm{2016} \\ $$$$\mathrm{If}\:\left({a}_{\mathrm{1}} {a}_{\mathrm{2}} \ldots{a}_{\mathrm{2016}} \right)^{\frac{\mathrm{1}}{\mathrm{2016}}} =\mathrm{2} \\ $$$$\mathrm{then} \\ $$$$\left(\mathrm{1}+{a}_{\mathrm{1}} \right)\left(\mathrm{1}+{a}_{\mathrm{2}} \right)\ldots\left(\mathrm{1}+{a}_{\mathrm{2016}} \right)\geqslant……
Question Number 55905 by gunawan last updated on 06/Mar/19 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left({x}_{\mathrm{2}{n}} +{x}_{\mathrm{2}{n}+\mathrm{1}} \right)=\mathrm{315} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left({x}_{\mathrm{2}{n}} +{x}_{\mathrm{2}{n}−\mathrm{1}} \right)=\mathrm{2016} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}_{\mathrm{2}{n}} }{{x}_{\mathrm{2}{n}+\mathrm{1}} }=… \\ $$…
Question Number 121410 by john santu last updated on 07/Nov/20 $$\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\sqrt[{\mathrm{p}}]{\mathrm{x}}−\mathrm{1}}{\:\sqrt[{\mathrm{q}}]{\mathrm{x}}−\mathrm{1}}\:?\: \\ $$ Answered by TANMAY PANACEA last updated on 07/Nov/20 $${x}={t}^{{pq}} \\ $$$${li}\underset{{t}\rightarrow\mathrm{1}}…
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Question Number 121302 by liberty last updated on 06/Nov/20 $$\:\left(\mathrm{1}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}−\mathrm{e}^{\mathrm{2x}} \right)\mathrm{sin}\:\left(\mathrm{3x}\right)}{\mid\mathrm{4x}\mid}\:?\: \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\lfloor\:\frac{\left(\mathrm{1}−\mathrm{e}^{\mathrm{2x}} \right)\mathrm{sin}\:\mathrm{3x}}{\mid\mathrm{4x}\mid}\:\rfloor\:? \\ $$ Answered by bemath last updated on 06/Nov/20…
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Question Number 121237 by benjo_mathlover last updated on 06/Nov/20 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+\mathrm{mx}\right)^{\mathrm{n}} −\left(\mathrm{1}+\mathrm{nx}\right)^{\mathrm{m}} }{\mathrm{x}^{\mathrm{2}} }\:=? \\ $$ Answered by liberty last updated on 06/Nov/20 $$\:\mathrm{L}'\mathrm{Hopital} \\…
Question Number 121222 by benjo_mathlover last updated on 06/Nov/20 $$\:\:\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\frac{\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }−\mathrm{x}+\mathrm{1}}{\mathrm{x}+\mathrm{1}}\:? \\ $$ Answered by liberty last updated on 06/Nov/20 $$\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\frac{\mid\mathrm{x}\mid\sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }}−\mathrm{x}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}}\right)}{\mathrm{x}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}}\right)}\:= \\…