Category: Limits

Question Number 55642 by gunawan last updated on 01/Mar/19 $$\mathrm{Prove}\mathrm{the}\mathrm{following}\mathrm{statements}:$$$$\mathrm{If}\mathrm{for}\mathrm{every}n,f_n\mathrm{form}\mathrm{ascend}\mathrm{function}$$$$\mathrm{and}\left\{f_n\right\}\mathrm{uniform}\mathrm{convergences}$$$$\mathrm{to}f\mathrm{at}[a,b],\mathrm{then}$$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\int_{{a}} ^{{b}} {f}_{{n}} \left({x}\right)\:{dx}\:\rightarrow\int_{{a}} ^{{b}}…

Question Number 55640 by gunawan last updated on 01/Mar/19 $$\mathrm{known}\mathrm{real}\mathrm{numbers}\mathrm{sequence}$$$$\left\{a_n\right\}\mathrm{and}\left\{b_n\right\}\mathrm{both}\mathrm{of}\mathrm{them}$$$$\mathrm{convergences}\mathrm{to}0.$$$$\mathrm{If}\left\{b_n\right\}\mathrm{monotonous}\mathrm{descend}$$$$\mathrm{and}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{a}_{{n}+\mathrm{1}} −{a}_{{n}} }{{b}_{{n}+\mathrm{1}} −{b}_{{n}}…

Question Number 55639 by gunawan last updated on 01/Mar/19 $$\mathrm{Known}a\in \mathbb{R}\mathrm{and}$$$$\mathrm{function}f:\mathbb{R}\to \mathbb{R}\mathrm{satiesfied}$$$$\mid xf\left(x\right)+a\mid <{\sin }^2(x-a).$$$$\mathrm{For}\mathrm{all}x\in \mathbb{R}$$$$\mathrm{value}\mathrm{of}\underset{x\to a}{\lim }f\left(x\right)..$$ Answered by kaivan.ahmadi…

Question Number 121172 by benjo_mathlover last updated on 05/Nov/20 $$\underset{x\to \mathrm{\infty }}{\lim }\frac{\sqrt{{\mathrm{x}}^2+1}-\mathrm{x}+1}{\mathrm{x}+1}?$$$$$$ Answered by TANMAY PANACEA last updated on 05/Nov/20 $$\underset{{x}\rightarrow\infty}…

Question Number 55634 by gunawan last updated on 01/Mar/19 $$\mathrm{If}\underset{x\to c}{\lim }\frac{a_0+a_1(x-c)+a_2{(x-c)}^2+\dots +a_n{(x-c)}^n}{{(x-c)}^n}=0$$$$\mathrm{then}\:{a}_{\mathrm{0}} +{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +..+{a}_{{n}} =.. \…