Discussion
For which of the following functions does the function take complex values arbitrarily close to any complex number, inside any arbitrary neighbourhood of 0 ?
Answer: B
The function is \(\sin(1/z)\).
The Casorati-Weierstrass theorem says that near an essential singularity a function takes values arbitrarily close to every complex number in every neighbourhood. \(\sin(1/z) = \sum (-1)^{k} z^{-(2k+1)}/(2k + 1)!\) has infinitely many negative powers in its Laurent series at 0, so 0 is an essential singularity.
\(e^{z}\) and \(\cosh z + \sinh^{2} z\) are entire, so they are bounded near 0 and stay close to their value at 0. \(z^{3} + z^{2} + 1/z\) has a simple pole at 0, so it tends to \(\infty\) there and cannot come close to every value.
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