Let $u(x,t)=f(x)\ast\left(\dfrac{1}{2\sqrt{\pi t}}e^{-\dfrac{(at+x)^2}{4t}}\right)$, and suppose that $f\in L^1$. Show that $$\lim_{t\rightarrow 0^+}\int_{-\infty}^\infty|u(x,t)-f(x)|dx=0$$
How can I prove this statement? $|u(x,t)-f(x)|$ is some ugly convolution, then we integrate over $\mathbb{R}$ and take the limit $t\rightarrow 0^+$.