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
Giuseppe
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.

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