Given functions $\alpha_t$ and $\beta_t$ from $[0,1]$ to $\mathbb C$ help me examine when this limit doesn't exist
If you assume both functions to be differentiable, then you can write $$\begin{split} \alpha_t &= 1+ \alpha \cdot t + o(t)\\ \beta_t &= \beta \cdot t + o(t) \end{split}$$ for some complex numbers $\alpha=\alpha_t^\prime(0)$ and $\beta=\beta_t^\prime(0)$. The ratio you are considering can be rewritten as $$\frac{\beta_t}{1-\alpha_t} = -\frac{\beta}{\alpha}+o(1)$$ and therefore, if you assume both $\alpha_t$ and $\beta_t$ to be differentiable at $t=0$, the limit will always exist, unless $\alpha^\prime(0)=\alpha=0$