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Question #a0310

1 Answer

Mar 30, 2017

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#

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