Why the delta at the end of the approximation? The equation to approximate an input signal with a unit impulse in Continuous Time, is shown below, before we take the limit $\hat{x}(t)=\frac{lim}{\Delta\rightarrow0}\sum^{\infty}_{-\infty}x(k\Delta)\delta_\Delta(t-k\Delta)\Delta$ &lt;-- why is there a final $\Delta$ multiplying the $\delta_\Delta(t-k\Delta)$?
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.