text: Q:
  
  $L = \lim\limits_{z \to 1} \frac{1-z}{1-z^\ast}$ evaluates incorrectly
  
  I'm about 90% certain this is a Mathematica issue, not me making a silly mistake.
  I'm trying to evaluate
  $$L = \lim_{z \to 1} \frac{1-z}{1-z^\ast}.$$
  Naively entering
  Simplify[
    Limit[(1 - z)/(1 - Conjugate[z]), z -> 1] 
    ,
    Element[z, Complexes]
          ]
  
  Mathematica says $ L = 1 $. 
  However taking $z = x + i (1-x) , \, x \rightarrow 1 $ looks like it gives $L=-i$,
  while $ z = 1 + i y , \, y \rightarrow 0$ gives $L = -1$. I carefully verified the algebra of these last two in Mathematica. Together they imply the limit does not exist.
  What is the issue here?
  More interestingly, is there a general approach for troubleshooting where and why such things happen?
  
  A:
  
  From
  Trace[
   Limit[(1 - z)/(1 - Conjugate[z]), z -> 1],
   TraceInternal -> True]
  
  one may observe a couple of things: First, Mathematica substitutes z -> z + 1, essentially transforming the limit to
  Limit[z/Conjugate[z], z -> 0]
  
  Second, it is using the default direction Direction -> -1, which may be observed in the assumption found in the details of the trace, namely, 0 < z < 1/4096.  Indeed this second observation is consistent with the behavior described in the documentation for Limit:
  
  Limit[expr, x -> Subscript[x, 0]] uses the setting Direction -> Automatic, which determines the direction from assumptions that have been given, using Direction -> -1 as the default. For limit points at infinity, the direction is determined from the direction of the infinity.
  
  So indeed a limit of 1 is the correct documented behavior.
  
  