# Estimate bound state energy for shallow finite well

It is well-known that the finite square well always has a bound state, no matter how shallow it is. Suppose the finite square well is given by $$V(x) = -V_0$$ for $$-a\le x \le a$$. The standard result is that the ground state can be obtained by solving the following transcendental equation in $$z$$:

$$z \tan z = \sqrt{z_0^2 - z^2}$$

where $$z_0 := \frac{a}{\hbar}\sqrt{2mV_0}$$ and the ground state energy can be calculated as

$$E = -V_0 \left(1-\frac{z^2}{z_0^2}\right)$$

My problem concerns shallow wells, i.e. the limit where $$z_0\to 0$$. I read a claim that the ground state energy $$E$$ can be estimated by $$-z_0^2 V_0$$ but I am having trouble proving this. I know some form of series expansion and truncating the higher order terms should take place, but I don't see how to approach the transcendental equation with this idea (since both $$z$$ and $$z_0$$ will be small).

Is this estimate correct? If not, what is the estimated energy in terms of $$V_0$$ and $$z_0$$ in this limit?