It is easy. Observe that $$T[x,t] = \frac{f(x/s)}{s}$$ with $t = \sqrt{t}$ and $f(x) = T[x,1]$. Since 
$$\int_{\mathbb R} f(x) dx=1 $$ we also have $$\int_{\mathbb R} \frac{f(x/s)}{s} dx=1 $$ simply by means of a trivial change of variables, defining $z = x/s$ where $s>0$.
Now take a bounded continuous function $g : \mathbb R \to \mathbb C$, with the aforementioned change of variables we have
$$\int_{\mathbb R} \frac{f(x/s)}{s} g(x) dx= \int_{\mathbb R} \frac{f(x/s)}{s} g(sx/s) s dx/s = \int_{\mathbb R} f(z) g(sz) dz\:.$$
Therefore
$$\lim_{t\to 0^+} \int_{\mathbb R}T[x,t]g(x) = \lim_{s\to 0^+} \int_{\mathbb R} f(z) g(sz) dz = \int_{\mathbb R} f(z) g(0) dz = g(0)  \int_{\mathbb R} f(z) dz = g(0) 1 = g(0)\:. \tag{1}$$
The only crucial passage is 
 $$\lim_{s\to 0^+} \int_{\mathbb R} f(z) g(sz) dz = \int_{\mathbb R} \lim_{s\to 0^+} f(z)  g(sz) dz$$
As quite mild condition that guarantees the passage is that $g$ is bounded as already required  (as consequence of *Lebesgue's dominated convergence theorem*).  
I stress that (1) is one of the possible ways to rigorously state that 
$$T[x,0] = \delta(x)\:.$$