In what follows we used Maple to show that 0 is a removable singularity, pole, and essential singularity respectively.  Change the function in each case but retain the same type of singularity.

> f:='f': L:='L': s:='s': S:='S': z:='z': Z:='Z':

f := z -> (cos(z) -1)/z^2:

`f(z) ` = f(z);

S := series((cos(Z) -1), Z=0, 11)/Z^2:

`f(z) ` = subs(Z=z,S);

S := series(f(Z), Z=0, 11):

`f(z) ` = subs(Z=z,S);

> f:='f': L:='L': s:='s': S:='S': z:='z': Z:='Z':

f := z -> sin(z)/z^3:

`f(z) ` = f(z);

S := series(sin(Z), Z=0, 13)/Z^3:

`f(z) ` = subs(Z=z,S);

S := series(f(Z), Z=0, 13):

`f(z) ` = subs(Z=z,S);

> f:='f': L:='L': p:='p': s:='s': z:='z': Z:='Z':

f := z -> z^2*sin(1/z):

s := convert(series(sin(z), z=0, 12), polynom):

S := subs(z=1/Z,s):

p := z -> subs(Z=z, expand(Z^2 * S)):

`f(z) ` = f(z);

L[9](z) = p(z);