Question

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

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

Answer 1

`+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.