I have a function f(x) which I would like to see going up for very low values of x: from an asymptotic study, I expect it to go to Infinity for low x; however, I can see an elbow and many oscillations. Are they due to numerical precision? How can I fix it? The function f(x) of interest is fon(beta).
    
     firstk = -8;
     lastk = 0;
     fon = 0.05379*beta/IntegralExact;
     IntegrandON = 
     Simplify[Abs[
      1 - (1 - I)/2*Sqrt[(beta/2)]*
        Cosh[(1 - I)*z*
           Sqrt[(beta/2)]]/(Sinh[(1 - I)/2*Sqrt[(beta/2)]])] // 
     ComplexExpand, beta > 0]^2;
     IntegralExact = FullSimplify[Integrate[IntegrandON, {z, -1/2, 1/2}]];

     Show[Flatten[{LogLogPlot[{fon}, {beta, 10^firstk, 10^lastk}, 
    PlotLegends -> {"On-axis"}, PlotStyle -> Black, 
    PlotRange -> {{10^firstk, 10^lastk}, {10^-6, 10^(8)}}, 
    GridLines -> {{46}, All}]}], ImageSize -> Large,  
    FrameLabel -> {"\[Beta]", "\.08F(\[Beta])"}, Frame -> True]

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/x9SnV.png