Taking "intermediate" limits, e.g. ${cal X}to 0$ but ${cal X}/epsilon to infty $ as $epsilonto 0$ - *without* taking $epsilon to 0$ directly

I have an expression that I would like to take limits of but not in a conventional sense. Take for example, an expression like

This expression involves intermediate variables ${cal X(epsilon)},{cal Y(epsilon)}$ whose limits are dependent on how they behave as $epsilonto 0$.
I would like to take limits of the above in such a way that ${cal X},{cal Y}to 0$ but also that ${cal X}/epsilon,{cal Y}/epsilontoinfty$ as $epsilonto 0$. This will allow the exponential term to decay to zero, whilst any term involving only ${cal X}$ or ${cal Y}$ will go to zero. Strictly speaking, the first term inside the second pair of brackets, ${cal Y}/epsilon$, diverges as $epsilon to 0$ but upon the action of multiplying by the prefactor $epsilon$, it will decay to zero as $epsilonto0$.

The result I would like to obtain is $-4epsiloncsc^{2}theta$, which can be evaluated only on the knowledge of ${cal X(epsilon)},{cal Y(epsilon)}$ as described above. However, I don’t actually want to take limits as $epsilon to 0$ of the whole expression. I want the result to still involve terms in $epsilon$ given that the above is a term in an asymptotic expansion in $epsilon$. If I take $epsilonto0$ along with $mathcal{X}to 0$ and $mathcal{Y}to 0$, I will just get zero, which is no use to me.

I am stumped on how to implement this in Mathematica, even though I know exactly what calculation I am doing. Are there any elegant ways I can accomplish this?