For Cauchy's principal value, one defines the principle value as
\begin{gather} PV\int_a^b f(x) dx = \lim_{\epsilon \to 0^+} \left[ \int_a^{c-\epsilon}f(x) dx + \int_{c+\epsilon}^b f(x) dx \right] \end{gather}
where $a \leq c \leq b$. In this definition, the limit from the left and the right approaches $c$ at equal rates.
Is there any mathematical or physical justification for taking the limit at the same rate aside from the fact that it has been defined this way? For example, could I define a new principal value as
\begin{gather} PV\int_a^b f(x) dx = \lim_{\epsilon \to 0^+} \left[ \int_a^{c-\epsilon}f(x) dx + \int_{c+2\epsilon}^b f(x) dx \right] \end{gather}
so that the limit from the right approaches twice as fast? I would also be interested if anyone knows of any physical applications where the limit has been taken at different rates.