Calculating Limit with option Analytic->True and $Assumptions yields Series::cas

Question

Overview

I am using Limit with the option Analytic->True to check the finiteness of some expressions with generic functions and I use $Assumptions to supply some general assumptions about the variables involved. While calculating the limit of some diverent expressions, I see Messages of the type Series::cas which involve assumptions, that I did not specify.

Example

I've managed to boil down the expression to the following (divergent) example:

$Assumptions=Element[x,Reals]&&0<x<1&&Element[ep,Reals]&&-1/1000<ep<0;
foo=x^(-1-ep)*(j[x]*nn[x,0]^(2-2*ep)
              +(nn[0,0]^(2-2*ep)-nn[0,y]^(2-2*ep)
               -(ff[x]*nn[x,0]^(2-2*ep))/ff[0]
               +(ff[x]*nn[x,y]^(2-2*ep))/ff[0]));
Limit[foo,x->0,Analytic->True]

The result is \[Infinity] j[0] nn[0, 0]^(2 - 2 ep) which is fine, since I expect the expression to be divergent.

The first evaluation of the Limit command yields the messages

Series::cas: "Warning: contradictory assumption(s) Re[x]>4096&&-(1/4096)<Im[x]<1/4096&&x\[Element]Reals&&0<x<1&&ep\[Element]Reals&&-(1/1000)<ep<0 encountered."
Series::cas: "Warning: contradictory assumption(s) Re[x]>4096&&-(1/4096)<Im[x]<1/4096&&x\[Element]Reals&&0<x<1&&ep\[Element]Reals&&-(1/1000)<ep<0 encountered."

What bothers me are the additional assumptions Re[x]>4096&&-(1/4096)<Im[x]<1/4096 which seem to be added inerternally by Series (which is at the heart of Limit if I understand things correctly).

Question

What exacly leads to these messages and how can I avoid them in general? I know that I can turn off messages using Quiet or more selectively using Off. I am more interested in getting to the root cause of this problem since I would like to use code based upon this to automatically detect divergent (sub-)expressions.

I found the somewhat related question Difficulty with computing a limit. However, the answer only addresses how to work around the problem in the specific case of the question.

Further observations

If I evaluate the command a second time there are no messages. After calling ClearSystemCache[] the messages reappar during the next evaluation. I suppose that the results of the call to Series are being cached.

If I supply the same assumptions to Limit directly via the option Assumptions, no messages appear:

ClearSystemCache[];
$Assumptions=True;
Limit[foo,x->0,Analytic->True,
      Assumptions->Element[x,Reals]&&0<x<1&&Element[ep,Reals]&&-1/1000<ep<0]

Using Assuming leads to messages, just like when using $Assumptions.

Further stripping down the example by removing subexpressions usually gets rid of the messages.

I've encountered this behaviour in Mathematica 11.0.1.0, 11.0.0.0, 10.2.0.0, 9.0.1.0 on Linux x86 (64bit).

Answer 1

Only for future reference: Apparently, the behaviour changed in newer versions of Mathematica.

In Mathematica 11.3.0.0 the messages disappeared completely, but the assumptions concerning the limit variable are silently ignored. In Mathematica 12.0.0.0 the assumptions on the limit variable are also ignored, but there is now a message about this:

Limit::alimv: Warning: Assumptions that involve the limit variable are ignored.

One can only influence the limit variable via the Direction option of Limit.