# Computing a tedious analytical limit

I would like to compute the following limit which I believe to exist for each $$\alpha,\beta>0$$. However, sometimes Mathematica is giving ComplexInfinity, sometimes no result at all. If I fix $$\alpha,\beta>0$$ it gives the correct result. Anyone knows a trick here in order to make Mathematica computing the limit?

g[x_, \[Alpha]_, \[Beta]_] =(1 - x)^-\[Beta] x^-\[Alpha] (-((
x^\[Alpha] HypergeometricPFQ[{\[Alpha], \[Alpha],
1 - \[Beta]}, {1 + \[Alpha], 1 + \[Alpha]}, x])/\[Alpha]^2) +
Beta[x, \[Alpha], \[Beta]] (Log[x] - PolyGamma[0, \[Alpha]] +
PolyGamma[0, \[Alpha] + \[Beta]]));

Limit[g[x, \[Alpha], \[Beta]], x -> 1, Direction -> "FromBelow",
Assumptions -> 0 < \[Alpha] && 0 < \[Beta]]


## 1 Answer

Making use of Series instead of Limit works to me in 13.2 on Windows 10:

g[x_, \[Alpha]_, \[Beta]_] =(1 - x)^-\[Beta] x^-\[Alpha] (-((x^\[Alpha]
HypergeometricPFQ[{\[Alpha], \[Alpha], 1 - \[Beta]}, {1 + \[Alpha], 1 + \[Alpha]}, x])/
\[Alpha]^2) +   Beta[x, \[Alpha], \[Beta]] (Log[x] - PolyGamma[0, \[Alpha]] +  PolyGamma[0, \[Alpha] + \[Beta]]));
(Series[g[x, \[Alpha], \[Beta]], {x, 1, 1},
Assumptions -> 0 < \[Alpha] && 0 < \[Beta] && x < 1] // Normal) /. x -> 1


(\[Pi] Csc[\[Pi] \[Beta]] (PolyGamma[0, \[Alpha]] - PolyGamma[0, \[Alpha] + \[Beta]]))/( Gamma[1 - \[Beta]] Gamma[1 + \[Beta]]) + 0^-\[Beta] (((\[Alpha] + \[Beta]) Gamma[1 + \[Alpha]]^2 Gamma[ 1 + \[Beta]] (PolyGamma[0, \[Alpha]] - PolyGamma[ 0, \[Alpha] + \[Beta]]))/(\[Alpha]^2 \[Beta] \ Gamma[\[Alpha]] Gamma[ 1 + \[Alpha] + \[Beta]]) + (\[Pi] Csc[\[Pi] \[Beta]] Gamma[ 1 + \[Alpha]] (-PolyGamma[0, \[Alpha]] + PolyGamma[0, \[Alpha] + \[Beta]]))/(\[Alpha] Gamma[ 1 - \[Beta]] Gamma[\[Alpha] + \[Beta]]))