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Answer 1 · 2017-03-06T14:58:58

#oo#

#lim_(x->oo) x(sqrt(x^2+2x)-2sqrt(x^2-x)+x)#

Take another #x# out of the bracketed terms:

#= lim_(x->oo) x^2 (sqrt(1+2/x)-2sqrt(1-1/x)+1)#

Now we do a Binomial series, #(1+x)^alpha = sum_(k=0)^(oo) ((alpha),(k)) x^k#, for the radicals:

#= lim_(x->oo) x^2 ((1+1/2* 2/x + mathcal O (1/x)^2)-2(1-1/2* 1/x + mathcal O (1/x)^2)+1)#

#= lim_(x->oo) x^2 (2/x + mathcal O (1/x)^2)#

#= lim_(x->oo) 2x + mathcal O (1/x)^0 = oo#