My handbooks "says" that a limit must be calculated for each part of the improper integral. When using the same limit for all parts, it is called the Cauchy Principal Value. My question is, what the difference is whether you use one limit for all parts or a limit for each part of the improper integral?
The improper integral has been defined as follows: $$\int _{a}^{c}\!f \left( x \right) {dx}$$ $$\int _{a}^{c}\!f \left( x \right) {dx}=\lim _{\epsilon\rightarrow 0} \left( \int _{a}^{b-\epsilon}\!f \left( x \right) {dx} \right) +\lim _{\mu\rightarrow 0} \left( \int _{b+\mu}^{c}\!f \left( x \right) {dx} \right) $$ And this is the CPV: $$ \int _{a}^{c}\!f \left( x \right) {dx}=\lim _{\epsilon\rightarrow 0} \left( \int _{a}^{b-\epsilon}\!f \left( x \right) {dx}+\int _{b+ \epsilon}^{c}\!f \left( x \right) {dx} \right) $$
There are cases in which the improper integral does not exist while the CPV still may exist. For example, take a look at the function
$$f(x)=\frac 1x$$
on the interval $[-1, 1]$.