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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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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
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@Nick Then you are almost done - what does $\overline {wz}=-wz$ imply? –  process91 Dec 16 '12 at 18:10
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Thanks for your help. –  Nickolas Dec 16 '12 at 18:31
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@Nick If you've figured it out, feel free to post an answer (and accept it). –  process91 Dec 17 '12 at 0:21

1 Answer 1

  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).

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