In formally proving that lim x → 1 ( x^ 3 − x^ 2 + 2 x ) = 2 , let ε > 0 be arbitrary. Choose δ = min ( ε/m , 1 ) . Determine m .

Question

i know how to do these problems i just don't know how to do it when you factor it out and it equals x(x^2-x+2)

Answer 1

Start with | f(x) - L |< ε --> |x^3 - x^2 +2x -2 | = |(x-1).(x^2+2)|<ε in other words |x-1| < ε/|x^2 + 2|. On the other hand since |x^2 + 2| = x^2 +2 > 2 for all x, we have ε/|x^2 + 2| < ε/2. Therefore it suffices to have δ=min(ε/2,1) or m=2.