intersection – Almost-crossing point of two curves when closer than a given precision

I’ve got two curves which asymptotically approach one another, say a(x) and b(x) (both are defined via NIntegrate, so I don’t know their analitic expressions).
I’d like to locate numerically the abscissa x_0 where their difference becomes less than a given error “err=0.001”.

In the image there’s an example of what I’m dealing with.
enter image description here

Is there a way to avoid writing a “For” cycle with many “If” conditions, which would be quite slow?