ca.analysis and odes - Integral determines function behaviour

Let us define:

$f(t) = t^{-1} int_{mathbf{R}^{3}} Exp[-frac{x^2}{2t}] h(x) dx,$

for a real function h. What can I say about this function if I know that

$f(t) rightarrow 1$.

I think that the convergence implies that

$t rightarrow int_{B(t)} h(x) dx$ (where B(t) is a ball of radius t)

grows like $t^{1/2}$ but I do not know how prove it formally. Moreover I have a feeling that this question is so simple (at least in formulation) that must have been answered somewhere.

(my question seems to be analitic, I put it also on the probability forum as may be it is possible to use some tools for Gaussian distribution/processes).