  We study the behavior of the critical price of an American put option near
maturity in the exponential L\'evy model when the underlying stock pays
dividends at a continuous rate. In particular, we prove that, in situations
where the limit of the critical price is equal to the stock price, the rate of
convergence to the limit is linear if and only if the underlying L\'evy process
has finite variation. In the case of infinite variation, a variety of rates of
convergence can be observed: we prove that, when the negative part of the
L\'evy measure exhibits an $\alpha$-stable density near the origin, with
$1<\alpha<2$, the convergence rate is ruled by $\theta^{1/\alpha}|\ln
\theta|^{1-\frac{1}{\alpha}}$, where $\theta$ is time until maturity.
