Asymptotic expansion of $\exp(-\frac{1}{\epsilon^2+\epsilon^3})$

Question

I am trying to find the asymptotic approximation of $\exp(-\frac{1}{\epsilon^2+\epsilon^3})$.

I reformed it into $\frac{\exp(\frac{1}{\epsilon})}{\exp(\frac{1}{\epsilon^2})\exp(\frac{1}{1+\epsilon})}$ so I can now Taylor expand $\exp(\frac{1}{1+\epsilon})$ but haven't got any further with the other two.

I also wasn't able to derive the expansion using integration by parts because I can't evaluate the expression when $\epsilon$ is 0.

Any hints would be appreciated!

Answer 1

We have \begin{align*} \exp \left( { - \frac{1}{{\varepsilon ^2 + \varepsilon ^3 }}} \right) &= \exp \left( { - \frac{1}{{\varepsilon ^2 }}\frac{1}{{1 + \varepsilon }}} \right) = \exp \left( { - \frac{1}{{\varepsilon ^2 }}\left( {1 - \varepsilon + \varepsilon ^2 + \mathcal{O}(\varepsilon ^3 )} \right)} \right) \\ & = \exp \left( { - \frac{1}{{\varepsilon ^2 }} + \frac{1}{\varepsilon } - 1 + \mathcal{O}(\varepsilon )} \right) = \exp \left( { - \frac{1}{{\varepsilon ^2 }} + \frac{1}{\varepsilon } - 1} \right)(1 + \mathcal{O}(\varepsilon )), \end{align*} as $\varepsilon \to 0$.