3. For the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$, the focal distance is $2 c$. The line $l$ passes through the points $(a, 0)$ and $(0, b)$, and the sum of the distances from the point $(1,0)$ to the line $l$ and from the point $(-1,0)$ to the line $l$ is not less than $\frac{4}{5} c$. Then the range of the eccentricity of the hyperbola is $\qquad$.
3. $\left[\frac{\sqrt{5}}{2}, \sqrt{5}\right]$ Detailed explanation: $l: b x+a y-a b=0$, from the problem we get: $\frac{b(a-1)}{\sqrt{a^{2}+b^{2}}}+\frac{b(a+1)}{\sqrt{a^{2}+b^{2}}} \geqslant \frac{4}{5} c$ which means $\frac{2 a b}{c} \geqslant \frac{4}{5} c \Rightarrow 5 a \sqrt{c^{2}-a^{2}} \geqslant 2 c^{2} \Rightarrow 5 \sqrt{e^{2}-1} \geqslant 2 e^{2}$, i.e., $4 e^{4}-25 e^{2}+25 \leqslant 0 \Rightarrow \frac{\sqrt{5}}{2} \leqslant e \leqslant \sqrt{5}$.