Question

Question #a0310

Answer 1

See below.

Explanation:

`(sqrt(x^4-x)-x^2)((sqrt(x^4-x)+x^2)/(sqrt(x^4-x)+x^2))=(x^4-x-x^4)/(sqrt(x^4-x)+x^2)=-x/(sqrt(x^4-x)+x^2)=-x/(x^2(sqrt(1-1/x^3)+1))=`

`=-1/(x(sqrt(1-1/x^3)+1))`

Now calling `y=sqrt(x)`

`lim_(x->0)sqrt(x^4-x)-x^2=lim_(y->0)y(sqrt(y^6-1)-y^3) = 0`

and

`lim_(x->oo)sqrt(x^4-x)-x^2=lim_(x->oo)(-1/(x(sqrt(1-1/x^3)+1)))=0`