# What is the simplest way to do asymptotic matching for boundary layer problem

I know the solution to the Boundary layer problem is something like this:

$$y_{in}+y_{out}-overlap$$

I have the Inner expansion as:

$$Y_{0}\left( x\right) \sim 1+c\int _{0}^{x}e^{-9x^{\frac {4} {3}}}ds,$$

and the outer expansion as as:

$$y_{0}\left( x\right) \sim e^{\frac {1-x^{\frac {2} {3}}} {8}}.$$

The problem i have is finding $c$. I know you do asymptotic matching by:

$$\lim_{X\rightarrow\infty}Y_0(X)=\lim_{x\rightarrow0^+}y_0(x)$$

How to i find the constant $c$ which could be something like this:

$$C=\dfrac {4\sqrt {3}\left( e^{1 / 8}-1\right) } {\Gamma \left( \dfrac {3} {4}\right) }$$

(Teacher was going to fast to explain, i might have copied it down wrong). And then to put into form of a "uniform solution".