# limit of an expression including BesselK function

i want to calculate the limit of the following expression when 'w' tend to zero. I have used the Limit function, it takes a lot of time for running without any result. could you please help me how to do that? Thanks in advance

  Limit[1 - ((BesselK[m, w]^2)/(BesselK[m - 1, w]*BesselK[m + 1, w])), w -> 0];


p.s: I know the answer is 1/m, but I want to show this.

• The limit is given by $\frac{1}{m}, \text{for}\; m\in\mathbb{R}$. Feb 19, 2021 at 20:09
• @yarchik Thanks for your response. I know that. but I want to show this Feb 19, 2021 at 20:22
• With suitable assumptions we can get this for m>1. In:= ll = Limit[1 - BesselK[m, w]^2/(BesselK[-1 + m, w] BesselK[1 + m, w]), w -> 0, Direction -> "FromAbove", Assumptions -> m > 1] Out= 1 - Gamma[m]^2/(Gamma[-1 + m] Gamma[1 + m]) In:= Simplify[FunctionExpand[ll, Assumptions -> m > 1]] Out= 1/m Feb 19, 2021 at 22:03
• @Daniel Lichtblau Thanks a lot! Feb 19, 2021 at 23:23

Limit[Limit[Normal[Series[1 - ((BesselK[m, w]^2)/(BesselK[m - 1, w]*