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

$$f(z)=\frac{e^{iz}}{z^2+1}$$

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?

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

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