General form of a Fourier transform of a function based on some conditions

Question

I have a function, $f(x)$, which has the following 3 conditions-

1. $f(x)$ is an even function of $x$: $f(-x) = f(x)$
2. Vanishes for large values of $|x|$: $$ \lim_{x\rightarrow\pm\infty} f(x)=0 $$
3. For some given set of parameters, $x_0$ and $\tau$, in the limit these parameters go to zero, the function converges to a delta distribution, $$ \lim_{x_0\rightarrow 0 \\ \tau\rightarrow 0}f(x;x_0,\tau) = \delta(x) $$

Given these conditions, can I write a general form for the Fourier transform of $f(x)$.