Math Help - Cauchy Principal value

1. Cauchy Principal value

Im a little confused on evaluating an improper integral from negative infinity to infinity.
If I know that $\int^\infty_{-\infty}f$ converges to A, how do I show that the Cauchy Principle value,
$\lim_{a\to \infty}\int^a_{-a}f=A$??
I know $\int^\infty_{-\infty}fdx=\lim_{a\to -\infty}\int^0_{a}fdx + \lim_{b\to\infty}\int^b_{0}fdx=-\infty+\infty$
How does this imply that the CPV is A???

2. Originally Posted by dannyboycurtis
Im a little confused on evaluating an improper integral from negative infinity to infinity.
If I know that $\int^\infty_{-\infty}f$ converges to A, how do I show that the Cauchy Principle value,
$\lim_{a\to \infty}\int^a_{-a}f=A$??
I know $\int^\infty_{-\infty}fdx=\lim_{a\to -\infty}\int^0_{a}fdx + \lim_{b\to\infty}\int^b_{0}fdx=-\infty+\infty$
How does this imply that the CPV is A???
This doesn't make sense to me, if you know that your first integral converges then by definition your last two integrals must exist and be finite.

Edit: That being said $\lim_{a\rightarrow \infty} \int_{-a}^{a} f = \lim_{a\rightarrow \infty} \int_{-a}^{0} f + \lim_{a\rightarrow \infty} \int_{0}^{a} f =A$

3. how do I show that $\lim_{a\rightarrow \infty} \int_{-a}^{0} f + \lim_{a\rightarrow \infty} \int_{0}^{a} f =A$?