A population growing in a habitat with limited resources shows logistic growth curve.
For logistic growth
$\frac{d N}{d t}=r N\left(\frac{K-N}{K}\right)$
If $K = N$ then $\frac{ K - N }{ K }=0$
$\therefore $ the $\frac{ d N }{ dt }=0$,
the population reaches asymptote.