How to solve #lim(root(3)(x^2) -2root(3)(x)+1)/(x-1)^2# ?

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Answer 1 · 2017-12-11T16:07:40

#1/9#

#(root(3)(x^2) -2root(3)(x)+1)/(x-1)^2 = ((root(3)(x)-1)/(x-1))^2# now calling #y = root(3)(x)#

#((root(3)(x)-1)/(x-1))^2 equiv ((y-1)/(y^3-1))^2 = (1/(y^2+y+1))^2#

and finally

#lim_(x->1)(root(3)(x^2) -2root(3)(x)+1)/(x-1)^2 = lim_(y->1) (1/(y^2+y+1))^2 = 1/9#

NOTE

We were using the polynomial identity

#(x^n-1)/(x-1) = 1+x+x^2+ cdots+ x^(n-1)#