Correct Answer - Option 4 : Essential singularity
\(f\left( z \right) = \sin \left( {\frac{1}{z}} \right)\)
We know that,
\(\sin z = z - \frac{{{z^3}}}{{3!}} + \frac{{{z^5}}}{{5!}} \ldots\)
\(f\left( z \right) = \sin \left( {\frac{1}{z}} \right) = \frac{1}{z} - \frac{1}{{3!{z^3}}} + \frac{1}{{5!{z^5}}} \ldots\)
If the number of negative powers of (z - a) in the f(z) is negative, then z = a is called an essential singularity.
∴ f(z) has infinite negative powers of z = 0 and z = 0 is an essential singularity.
