How to prove this limit of Airy Function.

Question

I have no idea how to prove this limit

$$\lim_{x\rightarrow \infty }\exp\left ( \frac{2x^{3/2}}{3} \right )\sqrt[4]{x}\mathrm{Ai}\left ( x \right )=\frac{1}{2\sqrt{\pi }}$$ where $\mathrm{Ai}(x)$ is Airy Function

Answer 1

Hint

The asymptotics of Airy function (see here and here ) is given by $$\text{Ai}(x)=e^{-\frac{2 x^{3/2}}{3}} \left(\frac{\sqrt[4]{\frac{1}{x}}}{2 \sqrt{\pi }}-\frac{5 \left(\frac{1}{x}\right)^{7/4}}{96 \sqrt{\pi }}+O\left(\left(\frac{1}{x}\right)^{13/4}\right)\right)$$ So,

$$\exp\left ( \frac{2x^{3/2}}{3} \right )\sqrt[4]{x}\mathrm{Ai}\left ( x \right )=\frac{1}{2 \sqrt{\pi }}-\frac{5 \left(\frac{1}{x}\right)^{3/2}}{96 \sqrt{\pi }}+O\left(\left(\frac{1}{x}\right)^3\right)$$ which shows the limit and how it is approached.