= Solution
Let $S=T_b\wedge T_{-a}$ and $p=\mathbb P(T_b<T_{-a})$. Set $R=\max(a,b)$. The exit time $S$ from $(-a,b)$ is at most the exit time from $(-R,R)$, so part (ii) gives $\mathbb ES<\infty$. Continuity ensures that exactly one of the two different boundary levels is reached at $S$.
The stopped <local martingale> $e^{X_{t\wedge S}}$ lies in $[e^{-a},e^b]$, hence is a <uniformly integrable martingale>. The <optional sampling theorem>, justified at the terminal time also by <bounded convergence theorem>, gives
$$
1=\mathbb Ee^{X_S}=pe^b+(1-p)e^{-a}.
$$
Thus the <boundary hitting probability from a diffusion scale function> is
$$
\boxed{\mathbb P(T_b<T_{-a})
=\frac{1-e^{-a}}{e^b-e^{-a}}
=\frac{e^a-1}{e^{a+b}-1}.}
$$
For integer $n\to\infty$, the events $\{T_b<T_{-n}\}$ increase to $\{T_b<\infty\}$. Indeed, on any path with finite $T_b$, continuity makes the path bounded below on $[0,T_b]$, so some lower barrier $-n$ has not yet been hit. The reverse inclusion is immediate. Taking the limit in the displayed probability gives
$$
\boxed{\mathbb P(T_b<\infty)=e^{-b}\qquad(b>0).}
$$
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