What is Maximum extent, limiting reactant?

The maximum extent $\xi_{\max}$ is the largest value of $\xi$ for which no amount is negative: $\xi_{\max} = \min_{\nu_i < 0} n_{i,0}/|\nu_i|$. The reactant that reaches zero at $\xi_{\max}$ is the limiting reactant. When the extent can also be negative (the reverse reaction), $\xi_{\min} = -\min_{\nu_i > 0} n_{i,0}/\nu_i$.