# Finding value of a function at limit zero

I want to evaluate the below function at limit zero. I tried with a function which has double differentiation for which it is giving me a finite value. For triple differentiation it is consistently giving output as zero. I want to know if the function given below can be evaluated and if not what could be the reason.

RF[r_] := (1/r)0.200731 (-0.000343151 r SphericalBesselJ[3, 0.137662 r] -
0.0642014 r SphericalBesselJ[3, 0.205217 r] +
0.116823 r SphericalBesselJ[3, 0.269851 r] -
0.188628 r SphericalBesselJ[3, 0.333395 r] +
0.174251 r SphericalBesselJ[3, 0.3964 r] +
0.0371598 r SphericalBesselJ[3, 0.459094 r] -
0.124404 r SphericalBesselJ[3, 0.521592 r] -
0.0878417 r SphericalBesselJ[3, 0.583959 r] +
0.0341412 r SphericalBesselJ[3, 0.646234 r] +
0.110696 r SphericalBesselJ[3, 0.70844 r] +
0.0979234 r SphericalBesselJ[3, 0.770594 r] +
0.0284896 r SphericalBesselJ[3, 0.832709 r] -
0.048473 r SphericalBesselJ[3, 0.894793 r] -
0.0981621 r SphericalBesselJ[3, 0.956851 r] -
0.11019 r SphericalBesselJ[3, 1.01889 r] -
0.0901751 r SphericalBesselJ[3, 1.08091 r] -
0.0513123 r SphericalBesselJ[3, 1.14292 r] -
0.00632141 r SphericalBesselJ[3, 1.20491 r] +
0.0352401 r SphericalBesselJ[3, 1.26689 r] +
0.0682566 r SphericalBesselJ[3, 1.32887 r] +
0.0908501 r SphericalBesselJ[3, 1.39084 r] +
0.103587 r SphericalBesselJ[3, 1.4528 r] +
0.108074 r SphericalBesselJ[3, 1.51475 r] +
0.106441 r SphericalBesselJ[3, 1.5767 r] +
0.100633 r SphericalBesselJ[3, 1.63865 r] +
0.0923663 r SphericalBesselJ[3, 1.70059 r] +
0.0828932 r SphericalBesselJ[3, 1.76253 r] +
0.0731573 r SphericalBesselJ[3, 1.82446 r] +
0.0637315 r SphericalBesselJ[3, 1.88639 r] +
0.0549955 r SphericalBesselJ[3, 1.94832 r] +
0.04711 r SphericalBesselJ[3, 2.01025 r] +
0.0401549 r SphericalBesselJ[3, 2.07217 r] +
0.0340995 r SphericalBesselJ[3, 2.13409 r] +
0.0288995 r SphericalBesselJ[3, 2.19601 r] +
0.02446 r SphericalBesselJ[3, 2.25793 r] +
0.0207028 r SphericalBesselJ[3, 2.31985 r] +
0.0175275 r SphericalBesselJ[3, 2.38176 r] +
0.0148598 r SphericalBesselJ[3, 2.44368 r] +
0.0126148 r SphericalBesselJ[3, 2.50559 r] +
0.0107338 r SphericalBesselJ[3, 2.5675 r] +
0.0091515 r SphericalBesselJ[3, 2.62941 r] +
0.0078254 r SphericalBesselJ[3, 2.69132 r] +
0.00670751 r SphericalBesselJ[3, 2.75323 r] +
0.00576856 r SphericalBesselJ[3, 2.81514 r] +
0.00497396 r SphericalBesselJ[3, 2.87705 r] +
0.00430427 r SphericalBesselJ[3, 2.93895 r] +
0.00373467 r SphericalBesselJ[3, 3.00086 r] +
0.00325262 r SphericalBesselJ[3, 3.06276 r] +
0.00284023 r SphericalBesselJ[3, 3.12467 r] +
0.00248963 r SphericalBesselJ[3, 3.18657 r] +
0.00218777 r SphericalBesselJ[3, 3.24848 r] +
0.00192993 r SphericalBesselJ[3, 3.31038 r] +
0.00170643 r SphericalBesselJ[3, 3.37228 r] +
0.00151461 r SphericalBesselJ[3, 3.43418 r] +
0.00134719 r SphericalBesselJ[3, 3.49608 r] +
0.00120281 r SphericalBesselJ[3, 3.55798 r] +
0.0010759 r SphericalBesselJ[3, 3.61989 r] +
0.000965963 r SphericalBesselJ[3, 3.68179 r] +
0.000868629 r SphericalBesselJ[3, 3.74369 r] +
0.000783953 r SphericalBesselJ[3, 3.80558 r] +
0.000708442 r SphericalBesselJ[3, 3.86748 r] +
0.000642491 r SphericalBesselJ[3, 3.92938 r] +
0.000583255 r SphericalBesselJ[3, 3.99128 r] +
0.000531332 r SphericalBesselJ[3, 4.05318 r] +
0.000484364 r SphericalBesselJ[3, 4.11508 r] +
0.000443062 r SphericalBesselJ[3, 4.17698 r] +
0.000405441 r SphericalBesselJ[3, 4.23887 r] +
0.000372266 r SphericalBesselJ[3, 4.30077 r] +
0.000341841 r SphericalBesselJ[3, 4.36267 r] +
0.000314949 r SphericalBesselJ[3, 4.42456 r] +
0.00029012 r SphericalBesselJ[3, 4.48646 r] +
0.000268132 r SphericalBesselJ[3, 4.54836 r] +
0.0002477 r SphericalBesselJ[3, 4.61025 r] +
0.000229577 r SphericalBesselJ[3, 4.67215 r] +
0.00021263 r SphericalBesselJ[3, 4.73405 r] +
0.000197583 r SphericalBesselJ[3, 4.79594 r] +
0.000183424 r SphericalBesselJ[3, 4.85784 r] +
0.000170843 r SphericalBesselJ[3, 4.91973 r] +
0.000158933 r SphericalBesselJ[3, 4.98163 r] +
0.000148347 r SphericalBesselJ[3, 5.04352 r] +
0.000138266 r SphericalBesselJ[3, 5.10542 r] +
0.000129305 r SphericalBesselJ[3, 5.16731 r] +
0.00012072 r SphericalBesselJ[3, 5.22921 r] +
0.000113092 r SphericalBesselJ[3, 5.2911 r] +
0.000105741 r SphericalBesselJ[3, 5.353 r] +
0.0000992116 r SphericalBesselJ[3, 5.41489 r] +
0.0000928812 r SphericalBesselJ[3, 5.47678 r] +
0.0000872617 r SphericalBesselJ[3, 5.53868 r] +
0.0000817795 r SphericalBesselJ[3, 5.60057 r] +
0.0000769142 r SphericalBesselJ[3, 5.66247 r] +
0.0000721356 r SphericalBesselJ[3, 5.72436 r] +
0.0000678925 r SphericalBesselJ[3, 5.78625 r] +
0.0000636919 r SphericalBesselJ[3, 5.84815 r] +
0.0000599513 r SphericalBesselJ[3, 5.91004 r] +
0.0000562072 r SphericalBesselJ[3, 5.97193 r] +
0.000052841 r SphericalBesselJ[3, 6.03383 r] +
0.0000494044 r SphericalBesselJ[3, 6.09572 r] +
0.0000462225 r SphericalBesselJ[3, 6.15761 r] +
0.0000428304 r SphericalBesselJ[3, 6.21951 r] +
0.0000395045 r SphericalBesselJ[3, 6.2814 r])


This is a wavefunction which includes spherical bessel functions

urF[r] := Abs[D[D[D[RF[r], r], r], r]]^2


I want to differentiate it three times and then evaluate at limit zero

DRF = Limit[urF[r], r -> 0, GenerateConditions -> False]


I am trying to evaluate the function at limit zero but it is giving me output as 0. I want a value even if it is very small. I tried using

$MinPrecision = 20; $MaxPrecision = 30;

But it is still not working. Any suggestions on how to overcome this. Thank you in advance

• what happens if you change urF[r] :=... to urF[r_] :=... and change urF8[r] to urF[r] as there is no urF8[r] anywhere in your code. Mar 12 at 6:22
• And you do not have to write D[D[D[RF[r], r], r], r] you can just write D[RF[r],{r,3}] Mar 12 at 6:25
• Yeah its urF[r] only. I will correct it in the original question. I was trying to see if it is differentiable that is why i wrote it like D[D[D[RF[r], r], r], r]. Mar 12 at 6:43
• if you remove Abs I get 0.0111299 as limit. Why did you add Abs ? Mar 12 at 6:45
• I wanted the function to have a positive value. I will check it once without absolute and see if it works for all of my other function and let you know. Thank you for your help good sir. Mar 12 at 6:57

After sometime, it does give answer with Abs also.
RF[r_] := (* see original post*)