How to show that the limit of cosh(z) as z goes to infinity does not exist?

Question

I am trying to understand what does it mean to calculate a limit of complex function as z approaches infinity. It is not intuitive to me. So far when calculating a finite limit (complex) it was clear that real part approaches real part and imaginary to imaginary. In an example exercise I was asked to calculate the limit of the function exp(z)/(exp(-z)+exp(z)) and the answer simplifying the function expression to 1/2cosh(z) concluding the limit of cosh(z) as z goes to infinity does not exist, and thus niether does the limit of the function, however without further explanation. I'm not quite sure how to practically show this conclusion. I would appreciate a detailed explanation/calculation. Thanks

Answer 1

Hint: Note that $\cos x=\cosh(ix)$. Now what happens to $\cos x$ as $x\to\infty$ along the $x$-axis?