I know that supremum is the lowest upper bound and infimum is the biggest lower bound. Also according to the supremum axiom, every non-nempty set bounded above has a supremum. Put
$$f(z) = \frac{|\frac{\sin(z)+\sin(1)-\sin(z+1)}{2-2\cos(1)}|-\frac{1}{2}\sinh(|z|)-\cosh(|z|)+1 }{\frac{1}{\pi^2-1}\cosh(\pi|z|)-\frac{1}{\pi^2-1}-\cosh(|z|)+1},\;\;z\in\mathbb{C}\setminus\{0\}.$$
I know that $f(z)$ is bounded above for every $z\in\mathbb{C}\setminus\{0\}$. Now my question is finding the supremum of $f(z)$.To this end I tried to get limit but I could not succeed. I appreciate any help to this matter.