What's the limit of a modified Bessel function of the first kind as the order approaches infinity?

Question

I'm trying to find an estimate of the cumulative distribution function of the Von Mises distribution.

The CDF is $D(x) = \frac{1}{2\pi I_0(b)}\left(xI_0(b)+2\sum_{j=1}^\inf \frac{I_j(b)\sin\left[j(x-a)\right]}{j}\right)$, where $I_j$ is the modified Bessel function of the first kind of order $j$.

This substantially exceeds my calculus skills. Assistance would be appreciated.

(My goal is actually to estimate the portion of the distribution between two points $x_1$ and $x_2$, if that makes it any easier.)

(Note: I'm aware of the questions asked regarding the limit of the Bessel function $I_j(x)$ as $x$ approaches infinity. My question concerns the limit when the order, $j$, approaches infinity.)

Answer 1

If you look here, you will see that $$I_n(x)\sim \frac 1{\sqrt{2 \pi n}} \left(\frac{ex}{2n} \right)^n$$