I am trying to understand what does it mean to calculate a limit of complex function as z approaches infinity. It is not intuitive to me. So far when calculating a finite limit (complex) it was clear that real part approaches real part and imaginary to imaginary. In an example exercise I was asked to calculate the limit of the function exp(z)/(exp(-z)+exp(z)) and the answer simplifying the function expression to 1/2cosh(z) concluding the limit of cosh(z) as z goes to infinity does not exist, and thus niether does the limit of the function, however without further explanation. I'm not quite sure how to practically show this conclusion. I would appreciate a detailed explanation/calculation. Thanks

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    $\begingroup$ The Question's wording and its tag for complex analysis suggests you might be looking for a limit at infinity in the sense of the complex plane. The only entire functions that have limits at infinity are polynomials, and among these, only constants have finite limits. $\endgroup$ – hardmath Jul 29 '16 at 20:35
  • $\begingroup$ Thanks. What if i write coshz as taylor polynomial ? $\endgroup$ – breeze Jul 29 '16 at 21:09
  • $\begingroup$ While $\cosh (z)$ is easily expressed as a power series related to that for $\cos (z)$ as the Accepted Answer mentions, this does not give a polynomial expression. $\endgroup$ – hardmath Jul 29 '16 at 21:17
  • $\begingroup$ Thank you. May I ask, Is there a theorem stating the above conclusion that only entire functions have limits at infinity are polynomials? Please share $\endgroup$ – breeze Jul 29 '16 at 21:38
  • $\begingroup$ en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis) $\endgroup$ – Hurkyl Aug 15 '16 at 18:24

Hint: Note that $\cos x=\cosh(ix)$. Now what happens to $\cos x$ as $x\to\infty$ along the $x$-axis?

  • $\begingroup$ Thank you Andre very nice observation. $\endgroup$ – breeze Jul 29 '16 at 20:46
  • $\begingroup$ There is no need for me writing a 15 sentence long answer, when I see something like this...+1 $\endgroup$ – imranfat Jul 29 '16 at 20:46
  • $\begingroup$ It is often not easy to see this. I tried calculating limits at different trajectories. What would you say is an efficient general approach to proving/disproving existance of limit at infinity? $\endgroup$ – breeze Jul 29 '16 at 20:57
  • $\begingroup$ Well, there are general criteria, as mentioned by hardmath, but those usually come later. At this time, all you can do is exploit your knowledge of standard functions, and get lucky. However, I think the above solution is fairly natural, we are travelling on the imaginary axis. $\endgroup$ – André Nicolas Jul 29 '16 at 21:04

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