confusion about poles and zeros of Lead compensator?
You're right, the given pole and zero are wrong. They should be

$$s_0=-\frac{1}{\tau}$$

and

$$s_{\infty}=-\frac{1}{\beta\tau}$$

because for $s=s_0$ the numerator becomes zero, and for $s=s_{\infty}$ the denominator becomes zero.