Question

Demonstrati egalitatea:


*va rog cat mai detaliat posibil

Answer 1

Il scrii pe 4x = ln e^4x si ai de aratat ca:

lim(x->oo) ln(e^4x) - ln(x^2+1) = oo

lim(x->oo) ln (e^4x)/(x^2+1) =oo

Aplici Teorema Stolz-Cesaro:

lim(x->oo) ln (e^4(x+1) - e^4x) /(x+1)^2 + 1 - x^2 - 1) = oo

lim(x->oo) ln [e^4x(e^4-1) / x^2 + 2x +1 - x^2)] =oo

lim(x->oo) ln [e^4x(e^4-1) / 2x+1] =oo

Aplici din nou Teorema Stolz-Cesaro:

lim(x->oo) ln {[ e^4(x+1) * (e^4-1) - e^4x * (e^4-1)] / [2(x+1) + 1 - 2x -1]} =oo

lim(x->oo) ln[(e^4-1)(e^(4x+4) - e^4)] / (2x+2+1-2x-1) =oo

lim(x->oo) ln[(e^4-1)^2 * e^4x] / 2 = oo , (∀) x un numar complex