# complex numbers

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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By $a*b!=0$ you mean $a*b\neq 0$, correct? The LaTeX for $\neq$ is "\neq" – icurays1 Dec 16 '12 at 17:40
What have you tried? How much are you having trouble with? – Mario Carneiro Dec 16 '12 at 17:53
@Nick Then you are almost done - what does $\overline {wz}=-wz$ imply? – process91 Dec 16 '12 at 18:10
Thanks for your help. – Nickolas Dec 16 '12 at 18:31
@Nick If you've figured it out, feel free to post an answer (and accept it). – process91 Dec 17 '12 at 0:21

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).