How should I approach a problem like this, given a function:


I need to find whether the limit of $|\frac{e^{iz}}{z^2+1}|$ at $i$ converges to some number, infinity, or if it does not converge to infinity or any number (IE.the real arguments converge to say 0 from negative side and infinity otherwise).

What would be the most useful things to try here?


closed as unclear what you're asking by Did, Leucippus, user10354138, Rebellos, rtybase Dec 12 '18 at 9:34

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  • $\begingroup$ You should use the fact that $f(z)$ you described is continuous at $i$. $\endgroup$ – NL1992 Dec 12 '18 at 0:03
  • 2
    $\begingroup$ Plug in first, does $z=i$ look like a special point? $\endgroup$ – Michael Burr Dec 12 '18 at 0:04
  • $\begingroup$ @MichaelBurr Sorry, a typo. $\endgroup$ – Dole Dec 12 '18 at 0:20
  • $\begingroup$ Don't understand the problem here. Plugging in (or taking the limit) $z=i $ is the obvious thing to try right? $\endgroup$ – Winther Dec 12 '18 at 1:00
  • $\begingroup$ @Winther Yes it goes to infinity, but does it go to somewhere else like zero as well approached from a different direction? To me it seems that it does given I think it might lead to a contradiction if i was a pole. $\endgroup$ – Dole Dec 12 '18 at 1:02

$|f(z)|=\frac {|e^{iz}|} {|z+i| |z-i|} \to \infty$ as $z \to i$.


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