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Discuss the singularities including the points at $\infty$: enter image description here

$\lim_{z\to 0} {z\over \sin z}=1$, so removable singularity, I dont know about $\infty$

$z\cos 1/z= z-{1\over 2! z}+{1\over 4!z^3}\dots$ so Essential Singularity at $0$

(iii) is also essential singularity at $0$

(iv) have no idea how to deal.

$16$ (i) poles at all points where $\sin z$ is $0$

(ii) essential singularity

(iii) pole of order $2$

(iv) essential singularity

17) (i) poles

(ii) essential

(iii) poles

(iv) poles

am I right ?

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It is not a complete answer . But in case of option 4, since neither of the limits $\lim_{z \to 0} \frac{1}{\cos(1/z)}$ and $\lim_{z \to 0} \cos(1/z)$ exist, the singularity is of essential type.

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