How I resolve this limit without use of De L'Hopital rules?

lim_(x->+oo)(sqrtxsin(1/(x+x^4)))/(xlog(1+e^(-2x))

1 Answer
Jun 28, 2018

+oo

Explanation:

I solved using the noticeable limits:

color(red)(lim_(x->0)(senx)/x = 1) and color(blue)(lim_(x->0)(log(1+x))/x = 1)

Because for x->+oo

  • 1/(x+x^4)->0
  • e^(-2x)->0

So, my limit will be:

lim_(x->+oo)((sqrtxcolor(red)sin(1/(x+x^4))1/(x+x^4))/color(red)(1/(x+x^4)))/((xcolor(blue)log(1+e^(-2x))(e^(-2x)))/color(blue)(e^(-2x)))=lim_(x->+oo)(sqrtx/(x+x^4))/(1/e^(2x))=lim_(x->+oo)(e^(2x)sqrtx)/(x+x^4) = +oo

Because the denominator is slower than numerator.