An example well-defined use case is if you want to compute a harmonic mean, e.g.

x = 2/(1/a +1/b)

This is a form of average where you are giving increased importance to the smaller number. It frequently pops up in science/engineering, e.g. in hydrology when you are computing flow of water underground.

In this case, it's "obvious" that when e.g. a is zero, you want 1/a to be zero so you simply return b. There's typically a clear, big distinction between "smallest realistic number", for instance 1e-3, and a zero, whether actual zero or 1e-34.

For instance Numpy offers a nice way of doing this:

  def safeInv(a):
    tol = 1e-16
    return np.division(1.0,a,out=np.zeros_like(a),where=np.where(a>tol))

This tells numpy to put the result of division into an array of zeros shaped like a, but only do the divide where a>tol, so it returns 0 where a<tol.

In this case, though, the programmer understands this division is somehow special and should allow div-by-zero, so it is handled specially.