Series -- one assumption works but a slight variation does not

Question

Question

Consider the following two lines of code, which differ only by replacing 1 by y:

SeriesCoefficient[
  (Exp[x + 1 + a] - 1)/(x + 1 + a), {a, 0, 0}, Assumptions -> x + 1 == 0]

SeriesCoefficient[
  (Exp[x + y + a] - 1)/(x + y + a), {a, 0, 0}, Assumptions -> x + y == 0]

The first version yields 1, which is the correct answer (just the limit of the function as "a" goes to zero). The second version yields:

(-1 + Exp[x + y])/(x + y)

which will cause problems for me later because x + y == 0.

Assuming x and y are real does not change the result.

Background

I want to take the limit of a large number of continuous functions of several variables:

Limit[
  Limit[Limit[Limit[f[y1, y2, y3, y4], y1 -> x1], y2 -> x2], y3 -> x3],
  y4 -> x4]

where each function f is similar to the one used above (ie it naively seems to have singularities but is actually continuous). I obtain much faster results using SeriesCoefficient:

SeriesCoefficient[f[y1 + a, y2 + a, y3 + a, y4 + a], {a, 0, 0}]

I want to be able to evaluate this with assumptions. For example:

SeriesCoefficient[
  f[x1 + a, x2 + a, x3 + a, x4 + a], {a, 0, 0}, 
  Assumptions -> x1 + x2 + x3 + x4 == 0]

but the assumptions are ignored.

Answer 1

This is not an answer to the question that was asked but it is too long for a comment.

My impression is that the question as stated is not reflecting the underlying desired result-- maybe I'm wrong but here goes. Is the idea to get the expansion, to low order, of (Exp[x + y] - 1)/(x + y) with both variables at the origin? If so there are (at least) two standard ways to go about this.

(1) Iterate Series over variables.

Series[Series[ee, {x, 0, 1}], {y, 0, 1}]

(* Out[168]= SeriesData[x, 0, {SeriesData[y, 0, {1, Rational[1, 2]}, 0, 2, 1], SeriesData[y, 0, {Rational[1, 2], Rational[1, 3]}, 0, 2, 1]}, 0, 2, 1] *)

(2) Make a new function of a new variable, effectively a "shrink factor" for the original variables, and expand at the origin.

Series[ee /. Thread[{x, y} -> t*{x, y}], {t, 0, 1}]

(* Out[169]= SeriesData[t, 0, {1, Rational[1, 2] (x + y)}, 0, 2, 1] *)

On a related note, there is support for multivariate limits over the reals in the works.

Answer 2

I don't know how useful this is for you, but one idea is to replace the problematic denominator part with a Piecewise:

SeriesCoefficient[
    (Exp[x+y+a] - 1)/(Piecewise[{{x+y, x+y != 0}}, 0] + a),
    {a, 0, 0},
    Assumptions -> x+y == 0
]

Simplify[%, x+y == 0]

E^(x + y)

1