Finding the limit of the following functions involving the Bessel function of the first kind

Question

I want to find the value of the functions $f[x,y,\theta]$ and $g[x,y,\theta]$ involving the Bessel function of the first kind, a Gaussian and a linear/algebraic function in the limit $\theta\rightarrow\frac{\pi}{2}$.

I am trying to work out the limits of

f[x_, y_, θ_] := (0.835817 E^(I x + I y - 1.25119 (x + y)^2) 
BesselJ[0, Sqrt[-y^2 + 400Cos[θ]]Sqrt[-2 + 2 Sec[θ]]] 
(2.50239 (x + y)-I Sqrt[Cos[θ]]))/Sqrt[Cos[θ]]

g[x_, y_, θ_] := (0.835817 E^(I x + I y - 1.25119 (x + y)^2) y 
BesselJ[1, Sqrt[-y^2 + 400 Cos[θ]]
Sqrt[-2 + 2 Sec[θ]]] Sqrt[-2 + 2 Sec[θ]])/(Sqrt[
Cos[θ]] Sqrt[-y^2 + 400 Cos[θ]])

On using the commands,

Limit[f[x, y, θ], θ -> π/2]

and

Limit[g[x, y, θ], θ -> π/2]

Mathematica doesn't evaluate the limits. Since this is a part of a theoretical problem (in physics) I'm working on, I should be able to evaluate these limits perfectly so that it conforms to the existing (established) results otherwise it should imply that I made a mistake while arriving at the specific forms of $f[x,y,\theta]$ and $g[x,y,\theta]$.

Is there a way to compute this?

Answer 1

Try this:

Series[f[x, y, θ], {θ, π/2, 0}]//Normal

I do not show the result, since it is much too long.

Have fun!