Maximization by imposing an asymptote constraint

I am trying to plot the profit curve for optimal e(or effort)
with respect to c.

Profit[c_, e_, F_] := 2 e + c e^2 - (2 e^2)/(1 - e) - 2 e B[F, e]

B[F_, e_] := 1/(1 - 2 e) (e/(1 - e) - e F )

effort[c_, F_] := 
 FindMaximum[{2 e + c (e)^2 - (2 e^2)/(1 - e) - 2 e B[F, e], 
    0 <= e < 1}, e][[2, 1, 2]]

Manipulate[
 Plot[{effort[c, F][[1]]}, {c, 0, 1}, PlotRange -> {0, 1}, 
  AxesLabel -> {c, Profit}, Exclusions -> {e == 0.5}, 
  ExclusionsStyle -> Directive[Red, Dashed]], {B, 0, 5}, {F, 0, 5}]

The problem is that the function I am trying to maximize has an asymptote at e=0.5 and I don’t know how to plot it.