I am lost with the following exercise that is posed in Steve Rosenberg's book "The Laplacian on a Riemannian Manifold":
Show that for a continuous function f,
\begin{equation} \lim_{t \to 0} \quad \frac{1}{\sqrt{4 \pi t}} \, \int_\mathbb{R} \exp \left(-\frac{(x - y)^2}{4t}\right) f(y) \, dy = f(x) \end{equation}
I am not sure how to tackle this - using L'Hopital's Rule doesn't really simplify the limit .. any hint would be a huge help, many thanks !!