I gather that $\displaystyle u(t)$ is the unit step function $\displaystyle u(t) = 0 \text{ for } t < 0,\ u(t) = 1 \text{ for } t \ge 0.$

From

this, you must show $\displaystyle \lim_{\epsilon \to 0} \delta_{\epsilon} (t) = 0 \text{ for } t \ne 0$ and $\displaystyle \int_{-\infty}^{\infty} \delta_{\epsilon} (t) dt = 1.$

So given $\displaystyle u(t) = 0 \text{ for } t < 0$, you just have to show $\displaystyle \lim_{\epsilon \to 0} \frac{e^{-t/\epsilon}}{\epsilon} (t) = 0 \text{ for } t > 0$ and $\displaystyle \int_{0}^{\infty} \frac{e^{-t/\epsilon}}{\epsilon} (t)\ dt = 1.$