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How to find the limit of such a complex function? $$ \lim_{z\rightarrow \infty} \frac{z \left| z \right| - 3 \Im z + i}{z \left| z \right|^2 +2z - 3i}. $$

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If you have an expression of the form $P(x)/Q(x)$ where $P$ and $P$ are polynomials, and the degree og $Q$ is larger than tjhe degree of $P$, then the limit when $x \rightarrow \infty$ is zero, by a simple, proof. Do that and try to adapt. – kjetil b halvorsen Sep 23 '12 at 15:48
up vote 2 down vote accepted

Consider moduli and use the triangular inequality.

The modulus of the numerator is at most $|z|^2+3|z|+1$ because $|\Im z|\leqslant|z|$ and $|\mathrm i|=1$. The modulus of the denominator is at least $|z|^3-2|z|-3$ because $|\mathrm i|=1$. Hence the limit of the ratio is $0$ when $|z|\to\infty$.

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