Question

Lim стремится к 1 x^3+x^2-3x+1/x^3-1

Answer 1

lim(x->1) (x^3 + x^2 - 3x + 1)/(x^3 - 1) =

x^3 + x^2 - 3x + 1 = x^3 - x^2 + 2x^2 - 2x - x + 1 = x^2(x - 1) + 2x(x - 1) - (x - 1) = (x - 1)(x^2 + 2x - 1)

x^3 - 1 = (x - 1)(x^2 + x + 1)

= lim(x->1) (x - 1)(x^2 + 2x - 1)/(x - 1)(x^2 + x + 1)  = lim(x->1) (x^2 + 2x - 1)/(x^2 + x + 1)  =  (1 + 2 - 1)/(1 + 1 + 1) = 2/3