How to omit extra vertical lines in Plot or Show

Question

How can I omit vertical lines in below output? They should not be present in the output and I don't know why does Mathematica show them. One reason could be: functions I am plotting are not Continuous functions. But even in this case, vertical lines shouldn't be present. This is the code which I am using:

r = 2;
d = 1;
dc = 1;
e = 2;
l = 1;
xr = 0;
z = -4;
dk = 1;
A = -1;
ro = 1;
v = 1;
g = 1;
xt = 0;
p1 = Plot[
    Sinh[
     1/2 Log[(1/
           4 (E^(-I t) + E^(I t))^2 (2 r - 3 e l ro + 
             1/2 A e (E^(-2 I t) + E^(2 I t)) l ro)^2 + 
          1/(dk l)  E^(I t))^2 (2 r - 3 e l ro + 
               1/2 A e (E^(-2 I t) + E^(2 I t)) l ro)^2 + 
            1/(dk l)
              4 (1/2 dk e (E^(-I t) - E^(I t))^2 l^2 r ro + 
                1/4 A dk e E^(-4 I t) (-1 + E^(2 I t))^2 (1 + E^(
                   4 I t)) l^2 r ro ]], {t, 0, 
    Pi}, PlotStyle -> ({Blue} & /@ {Large, Medium})];
Show[p1, p2]

Answer 1

Those are probably precision artifacts caused by a denominator in a complicated fraction. Let's define your function:

foo[t_?NumericQ] := 1/(4 l ro) I ...

and investigate it:

{plot, {data}} = Plot[
  foo[t], {t, .9, 1.2}, EvaluationMonitor :> Sow[{t, foo[t]}]
] // Reap;

Select[data, Positive@Chop@#[[2]] &]

{{1.03468, 0.236293}, {1.05918, 0.250562}, {1.16303, 0.303766}, {1.1694, 0.306664}, {1.16009, 0.302416}, {1.19671, 0.318612}, {1.15243, 0.298852}, {1.17089, 0.307335}, ...

Chop @ foo[1.0346814388488432`]

0.236293

Maybe simplification can help:

foo[t_?NumericQ] := Evaluate @ FullSimplify[1/(4 l ro) I ...

Plot[foo[t], {t, 0, Pi}, PlotRange -> All]

It did :)