Asymptotic behavior of $1/(a^2+\epsilon^2)$ as $\epsilon\to0$

Question

A limit that often arises in physics is $$ \lim_{\epsilon \to 0} \frac{ \epsilon }{ a^2 + \epsilon^2 } = \pi \delta(a) ............ (1) $$ Is there a similar sort of limit for $$ \lim_{\epsilon \to 0} \frac{ 1 }{ a^2 + \epsilon^2 } = \frac{1}{a^2} + ??? $$ For $a \neq 0$, this limit is well-defined and can easily be computed. For $a = 0$, it blows. Can this blow-up be described in terms of any delta function as in (1) of derivatives thereof?