3. $b \sqrt{a^{2}-b^{2}}$.

Make the transformation $(x, y) \rightarrow\left(x, \frac{a y}{b}\right)$, we know that $S_{\triangle A B F}$ reaches its maximum value if and only if $A B \perp O F$.
3. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0), F$ is the right focus of the ellipse, $AB$ is a chord passing through the center $O$. Then the maximum value of the area of $\triangle ABF$ is