information:highlighting, quality:correction
Cauchy principal value

Consider the difference in values of two limits:

The former is the Cauchy principal value of the otherwise ill-defined expression

Similarly, we have

but

The former is the principal value of the otherwise ill-defined expression

All of the above limits are cases of the indeterminate form ∞ − ∞ {\displaystyle \infty -\infty }.

These pathologies do not affect "Lebesgue-integrable" functions, that is, functions the integrals of whose absolute values are finite.