Consider the integral:
$$\int_0^\infty\sin(x)dx.\tag1$$
It's clearly divergent, but if we regularize it as
$$\int_0^\infty\sin(x)e^{-x/a}dx=\frac{a^2}{a^2+1},\tag2$$
we can take the limit of $a\to\infty$ and get some value (here $1$).
This reminds me of an approach similar to the following: a function, e.g. $x\mapsto e^{1/x}$, isn't defined at $x=0$. But if we take the limit $x\to0^-$, we get some value (here $0$). But if we take the limit from another side, we won't get the same value (actually we'll get either infinity (for $x\to0^+$), or no limit at all).
It seems that doing regularization $(2)$ and taking the limit $a\to\infty$ is very similar to approaching the integral's value from some "side" of function space on which it is defined. So I assume that it can, as the function discussed above, have different limits when approached from other "sides", or not have any. Is it true? What would be an example of such approach directions (i.e. regularizing functions $\xi(a)$), which would give some finite value different from $1$?