Question Number 55640 by gunawan last updated on 01/Mar/19
$$\mathrm{known}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{sequence} \\ $$$$\left\{{a}_{{n}} \right\}\:\mathrm{and}\:\left\{{b}_{{n}} \right\}\:\mathrm{both}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{convergences}\:\mathrm{to}\:\mathrm{0}. \\ $$$$\mathrm{If}\:\left\{{b}_{{n}} \right\}\:\mathrm{monotonous}\:\mathrm{descend} \\ $$$$\mathrm{and}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{a}_{{n}+\mathrm{1}} −{a}_{{n}} }{{b}_{{n}+\mathrm{1}} −{b}_{{n}} }\:. \\ $$$$\mathrm{then}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{a}_{{n}} }{\mathrm{2}{b}_{{n}} }=.. \\ $$