Setting numerator terms to zero only

Question

I have many expressions of the form $c_1/\epsilon, c_2/\epsilon \dots ,c_n/\epsilon$ where $n$ is a large number and I also have expressions of the form $b_1\epsilon, b_2\epsilon, \dots ,b_m \epsilon$ where $m$ is also very large. $c_i$ and $b_i$ are just $\epsilon$-independent variables or numbers.

I would like to put the terms in which $\epsilon$ appears in the numerator to zero without affecting the terms in which $\epsilon$ appears in the denominator. Is there a way to automate this in Mathematica? Probably the answer is yes and the solution is very simple but I have not found the correct syntax yet.

   expr/.eps->0

gives rise to errors because of producing indeterminate forms, as expected.

Thanks!

Answer 1

expr = 
  {(256 m2)/(ϵ M (-1 + b x) (1 + b x)), (256 ϵ m1)/(M^3 (-1 + b x) (1 + b x))};

Replace[#,Times[PatternSequence[___, Verbatim[ϵ]]] -> 0, {1}] & /@ expr
(* {(256 m2)/(M (-1 + b x) (1 + b x) ϵ), 0} *) 

Answer 2

It would be nice to obtain a more direct route but the following simple trick gets the job done:

   expr/.{1/eps->x};
   %/.{eps->0}
   %/.{x->1/eps}

Answer 3

You could use @DanielLichtblau's solution from the comments:

Normal @ Series[{b1 ϵ, b2 ϵ, c1/ϵ, b3 ϵ, c2/ϵ, c3/ϵ}, {ϵ, 0, 0}]

{0, 0, c1/ϵ, 0, c2/ϵ, c3/ϵ}

Answer 4

If the input expression expr only contains terms of the form 1/eps and eps, then you can use Coefficient:

Dot[Coefficient[expr, eps, {-1, 0}], {1/eps, 1}]