Im a little confused on evaluating an improper integral from negative infinity to infinity.

If I know that $\displaystyle \int^\infty_{-\infty}f$ converges to A, how do I show that the Cauchy Principle value,

$\displaystyle \lim_{a\to \infty}\int^a_{-a}f=A$??

I know $\displaystyle \int^\infty_{-\infty}fdx=\lim_{a\to -\infty}\int^0_{a}fdx + \lim_{b\to\infty}\int^b_{0}fdx=-\infty+\infty$

How does this imply that the CPV is A???