Show that $z=0$ is not a branch point for the function $f(z)= \frac{\sin\sqrt{z}}{\sqrt{z}}$. Is it a removable singularity?
Definition- a branch point of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point.
I have no idea how to prove the first part. For the second part, I am not sure if the $lim_{z\to 0}\frac{\sin z}{z} = 1$, can be applied and hence can we conclude if its a removable singularity? Because I think as $z\to 0$, $$f(z)\to \frac{\sin(e^{\frac{\iota\theta}{2}})}{e^{\frac{\iota\theta}{2}}}$$ would not exist. I have assumed $\sqrt{z}:=\frac{1}{2}(\log(|z|)+ \arg(z))$ where $0\leq \arg(z)<2\pi$
Any help is appreciated. Thanks.