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I know that $f$ is continuous on $[a,b]$ with $ab\neq0$, $f(a)f(b)\neq0$ and the complex numbers:

$ z = a^2 + f(a)i $ $ w = b^2 - f(b)i $

$|\bar w + z| = |w - \bar z|$

1)Prove that $w\cdot z$ is an imaginary number

2)Calculate the limit $\lim_{x\to\infty}\frac{f(a)x^3 - f(b)x + 5}{f(b)x^2 + f(a)x - 3}$

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    @Nick If you've figured it out, [feel free to post an answer (and accept it)](http://meta.math.stackexchange.com/questions/2637/policy-on-accepting-my-own-answer).2012-12-17

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  1. For any complex numbers $z,w$ we have $\operatorname{Re}(wz) = \frac12 (|\bar w+z|^2-| w-\bar z|^2)$

  2. The limit $\lim_{x\to\infty}\frac{f(a)x^3 - f(b)x + 5}{f(b)x^2 + f(a)x - 3} = \lim_{x\to\infty}\frac{f(a)x - f(b)/x^2 + 5/x^3}{f(b) + f(a)/x - 3/x^2} $ does not exist (as a finite number).