Dirac Delta Function

$$\delta (x)=\{\begin{array}{ll}0,& x\ne 0\\ \infty ,& x=0\end{array}=\underset{a\to 0}{\lim }\frac{1}{|a|\sqrt{\pi }}{e}^{-(x/a{)}^2}$$ $${\int }_{-\infty }^{\infty }\delta (x)dx=1$$ $${\int }_{-\infty }^{\infty }\delta (t-a)f(t)dt=f(a)$$