Oblicz granicę  Przykład w załączniku

Pierw zauważ że: [latex]frac{3n+4}{5n+2}=frac{5n+2-(2n-2)}{5n+2}=1-frac{2n-2}{5n+2}=1+frac{-2n+2}{5n+2}[/latex] Teraz liczymy granicę: [latex]lim_{n o infty} left(1+frac{-2n+2}{5n+2} ight)^{2n+3} = \=lim_{n o infty} left( left( left( 1+frac{-2n+2}{5n+2} ight)^{frac{5n+2}{-2n+2}} ight)^{frac{-2n+2}{5n+2}} ight)^{2n+3}= \ =lim_{n o infty} left( left( 1+frac{-2n+2}{5n+2} ight)^{frac{5n+2}{-2n+2}} ight)^{frac{(-2n+2)(2n+3)}{5n+2}} =lim_{n o infty} e^{frac{-4n^{2}-4n+6}{5n+2}}= \ =lim_{n o infty} e^{frac{-4n-4+frac{6}{n}}{5+frac{2}{n}}}=lim_{n o infty} e^{frac{-4n-4}{5}}=  \ [/latex] [latex]= [e^{-infty}]=0[/latex]