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I need to prove that a non constant complex tends to infinity when the argument tends to infinity. I'd be happy for a hint.

thanks

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    $x^n + a_{n-$1$} x^{n-1} + \cdots + a_1 x + a_$0$ = x^n(1+a_{n-1}/x+\cdots+a_1/x^{n-1}+a_0/x^n)$2012-11-07

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$\def\abs#1{\left|#1\right|}$*Hint*: You have by the triangle inequality $ \abs{\sum_{k=0}^n a_k z^k} \ge \abs{a_n}\abs{z}^n - \sum_{k=0}^{n-1}\abs{a_k}\abs z^k $

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If the polynomial is $f(z)$, consider $g(w)=1/f(1/w)$ and prove that $\displaystyle\lim_{w\to 0} g(w)=0$. This means that $\displaystyle\lim_{z\to \infty} f(z)=\infty$.

This is a standard technique for handling infinity: bring it to zero using the inversion $w=1/z$.