Why the delta at the end of the approximation?
Here the signal is in continuous time domain. We can approximate any signal with weighted integral of unit impulse. As the signal is in continuous time domain integration is used instead of summation.

The actual equation is ,
$$x(t)=\int_{-\infty}^{\infty}{x(t_0)\delta(t-t_0)dt}$$

In your equation ${lim}_{\Delta\rightarrow0}\sum^{\infty}_{-\infty}$ stands for integration so $\Delta$ is required at the end, which stands for dt in integral.

Hope you clear that.