= Solution
The <WKB approximation> fails at a simple <classical turning point> $x_t$, because $p(x_t)=0$ makes its amplitude singular and violates the <WKB validity condition>. Locally replace $V(x)-E$ by $V'(x_t)(x-x_t)$. For a forbidden region on the right, $V'(x_t)>0$ and the rescaled coordinate $z=[2mV'(x_t)/\hbar^2]^{1/3}(x-x_t)$ changes the time-independent <Schrödinger equation> into the <Airy ordinary differential equation> $\psi_{zz}-z\psi=0$. Reversing the coordinate handles a forbidden region on the left.
The two independent <Airy functions> have asymptotic forms, for $z,w>0$,
$$
\begin{aligned}
\operatorname{Ai}(z)&\sim\frac{z^{-1/4}}{2\sqrt\pi}e^{-2z^{3/2}/3},&\operatorname{Bi}(z)&\sim\frac{z^{-1/4}}{\sqrt\pi}e^{2z^{3/2}/3},\\
\operatorname{Ai}(-w)&\sim\frac{w^{-1/4}}{\sqrt\pi}\sin(2w^{3/2}/3+\pi/4),&\operatorname{Bi}(-w)&\sim\frac{w^{-1/4}}{\sqrt\pi}\cos(2w^{3/2}/3+\pi/4).
\end{aligned}
$$
Match these to the <WKB approximation> in an overlap region where the linearization of the <potential energy> and the large-argument <asymptotic expansions> both apply. If $K_t=\hbar^{-1}|\int_{x_t}^x\chi\,dx|$ on the forbidden side and $s_t=\hbar^{-1}|\int_{x_t}^xp\,dx|$ on the allowed side, the <Airy turning-point connection formula> is
$$
\boxed{\frac{B}{\sqrt\chi}e^{-K_t}\ \longleftrightarrow\ \frac{2B}{\sqrt p}\sin(s_t+\pi/4),\qquad\frac{A}{\sqrt\chi}e^{K_t}\ \longleftrightarrow\ \frac{A}{\sqrt p}\cos(s_t+\pi/4).}
$$
Thus a decaying exponential acquires a factor two in its allowed-side amplitude, whereas a growing exponential does not. The phase convention uses positive integrals measured away from the <classical turning point>, so it applies at both ends of the well without an extra orientation sign. These are leading semiclassical matching rules for simple, separated <classical turning points>.
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