I have this integrand given by tinteg[z,zm,zh] where I want to integrate it in the range [0,zh]. However, the integrand is divergent at zh so I added a small parameter eps away from zh so that the integral range is now [0,zh+eps]. Doing this, I can then take the principal value of the integral for some eps.

d = 3;
ag = 20;
pg = 20;
wp = 40;
f[z_, zh_] := 1 - (z/zh)^(d + 1);
tinteg[z_, zm_, zh_] := -1/(f[z, zh] Sqrt[1 - (zm^(2 d) f[z, zh])/(z^(2 d) f[zm, zh])])

th[eps_?NumericQ, zh_?NumericQ] := Module[{epsr, zhr}, {epsr, zhr} = Rationalize[{eps, zh}, 0]; NIntegrate[tinteg[z, zhr + epsr, zhr], {z, 0, zhr, zhr + epsr}, Method -> PrincipalValue, AccuracyGoal -> ag, PrecisionGoal -> pg, WorkingPrecision -> wp]]

th[10^-9, 100]
0.0003162277660196573967272197281906499

th[10^-10, 100]
NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in z near {z} = {99.9999999998213805278900613760597337931183240142657527684191416030996934771392010899316650}. NIntegrate obtained -57.3106917390753700387564367567637817259282112119845402535003746786552197747577990043682551 and 1.85452812514993194537323109011792300865115544959913889998941281161057638475694312252915879`90.*^-10 for the integral and error estimates.
0.0001000000000000889405834231780766118

th[10^-15, 100]
NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in z near {z} = {99.9999999999997866007398081727804784995174546220786282909845407978056563093393638396629520}. NIntegrate obtained -57.3107914228724344179090132179535225292646529296309563458710154094184035242065488421646708 and 0.000096232917814206821120246506287335370896029249848796293573094556559070111933328788951276428`90. for the integral and error estimates.
3.161570082363756029776335213343*10^-7

The problem is, as I take eps to be smaller and smaller it will reach a point where error occurs which is related to convergence issues. For eps=10^-9, I can still get a result without error. However, for eps=10^-10 error already occurs.

I have set the corresponding AccuracyGoal,PrecisionGoal,WorkingPrecision to try and go over this problem however it seems not to be working. I think the smallest eps that I might need is around eps=10^-15 in order to justify the accuracy of what I'm doing, so if I can just push the convergence to this limit then everything will be good.