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Using "Limit" when the limit is a delta function

I'm just wondering whether I can get Mathematica to understand the
following limit (I'm using version 6.01).

The standard solution for u(x,t) of the diffusion equation  u_t = D
u_xx   over an infinite domain with initial condition   u(x,0) =
delta(x)    (Dirac delta) is

u[x_,t_] = (1/Sqrt[4 Pi D t]) Exp[-x^2/(4 D t)]

This solution, in fact, defines a family of functions whose limit as
t-->0  yields the Dirac delta, as the initial condition dictates.  In
Mathematica I tried

Limit[u[x, t], t -> 0]

and got nothing from it.  Can I get Mathematica to handle this sort of
limit?  I realize that it involves an essential singularity at t=0,
but I would've guessed that Mathematica is equipped to deal with it.
An extra option needs to be specified or something?


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