Asymptote in a logistic growth curve is obtained when:

In the logistic growth equation
$$\frac{dN}{dt}=rN\left(\frac{K-N}{K}\right)$$
, where $K$ is carrying capacity and $N$ is population density. When $N$ reaches $K$, the term $\frac{K-N}{K}$ becomes zero, the growth rate $\frac{dN}{dt}$ becomes zero, and the curve reaches its maximum limit (asymptote).