Allow me to replace $k$ by $x$. The kernel
$$G(x,t)=(4\pi it)^{-1/2}e^{ix^2/4t}$$
is the Green function of the Schrödinger equation, which can be written in the integral form
$$G(x,t)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ikx}e^{-ik^2 t}dk.$$
In the limit $t\rightarrow 0$ we then have an integral representation of the delta function,
$$2\pi\delta(x)=\int_{-\infty}^\infty e^{-ikx}dk.$$
Both these integral representations need to be understood in the distributional sense (meaning that they need to be multiplied by a test function and integrated).