Hypergeometric2F1 in Mathematica

Question

I would like to see some limits and plots of the Hypergeometric2F1 function in Mathematica 13.0. I don't have experience with hypergeometric functions, but need to compute the following limit and plot the inverse of this function:

g[x_]:=Sqrt[(x^2 (2/3 - a + b))/(-(2/3) + a)]Hypergeometric2F1[1/2, 1/(2 + 3 a - 3 b), 1 + 1/(2 + 3 a - 3 b), -((f x^(2 + 3 a - 3 b) (2/3 - a + b))/(-(2/3) + a))]

where a,b and f are positive values and a>b.

 With[{a = 1.3, b = 0.7, f = 0.001}, 
 ContourPlot[Sqrt[(x^2 (2/3 - a + b))/(-(2/3) + a)]Hypergeometric2F1[1/2, 1/(2 + 3 a -3 b), 1 + 1/(2 + 3 a - 3 b), -((f x^(2 + 3 a - 3 b) (2/3 - a + b))/(-(2/3) + a))] == t, {t, 1, 2}, {x, 1, 2}, AspectRatio -> Automatic, FrameLabel -> {t, x}]]

Answer 1

  Block[{a = 1.3, b = 0.7, f = .01, ps = {1, 2}, eqns}, 
  eqns[p_] := (Sqrt[(x^2 (2/3 - a + b))/(-(2/3) + a)]) Hypergeometric2F1[1/2, 1/(2 + 3 a - 3 b), 1 + 1/(2 + 3 a - 3 b), -((f x^(2 + 3 a - 3 b) (2/3 - a + b))/(-(2/3) + a))] == p t;
  ContourPlot[Evaluate[eqns /@ ps], {t, 0.1, 5}, {x, 0.1, 100},AspectRatio -> 1, FrameLabel -> {t, x}, PlotLegends -> StringTemplate["p = ``"] /@ ps]]