I want to find:
$$\lim_{t\to\infty}X_{t}$$
where:
$$X_{t} = \frac{A_{t}^{a}}{B_{t}}$$
I know that:
$$\lim_{t\to\infty}A_{t}=0$$
and
$$\lim_{t\to\infty}B_{t}=0$$
Can I say with certainty that:
$$\lim_{t\to\infty}X_{t}=\frac{0^{a}}{0} = \left\{\begin{array}{l l} 0 & \text{if $a > 1$}\\ 1 & \text{if $a=1$}\\ \infty & \text{if $a<1$} \end{array} \right.$$
This is, can I compare dominance of expressions?
Note that I don't have closed-form expressions for $A(t)$ and $B(t)$ so I cannot do L'Hopital here. I do know their limit though, as given above.