Simplify $\log(\sinh z)$ when $|z|$ tends to $0$?

Question

I was given $\log(\sinh z)$ and I need to show it tends to $\log z$ when $|z|$ tends to $0$.

I have tried converting $z$ to $x+iy$ then split $\sinh z$ but that doesn't seem to get me anywhere.

I know $\log z=\log|z|+i{\rm Arg}(z)$ but I have no idea how to use this.

Any hint would be helpful thanks!

Answer 1

$\begin{array}\\ \sinh(z) &=\dfrac{e^z-e^{-z}}{2}\\ &\approx \dfrac{(1+z+z^2/2)-(1-z+z^2/2)}{2} \quad\text{(for small z)}\\ &=z \end{array} $

Now take $\ln$.