 Legendre boundary layer at x equals one

The endpoint Laplace method for a Legendre polynomial ceases to localize when $n\sqrt{x-1}=O(1)$. On the scale $x=1+\nu /n^2$, its real integral representation givesThe Modified Bessel function of the first kind equals one at $\nu =0$ and has large-$\nu$ behavior $e^{\sqrt{2\nu }}/\sqrt{2\pi \sqrt{2\nu }}$, matching the Hyperbolic Debye asymptotic for Legendre polynomials. This boundary layer resolves an increasingly steep function at an endpoint where its value remains fixed.