Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
1  
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

1 Answer 1

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 $|z|\to\infty$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.