A certain type of improper integral, due to Cauchy, commonly abbreviated to PV. For example, the integral ${\int }_{-\infty }^{\infty }\frac{x}{1+x^2}\text{d}x$ does not converge. There is infinite area above the x-axis and infinite area below it. The integral ${\int }_{-R}^S\frac{x}{1+x^2}\text{d}x$ does not have a limit, as R,S tend to infinity. However, the limit ${\text{lim}}_{R\to \infty }{\int }_{-R}^R\frac{x}{1+x^2}\text{d}x$ does exist, equalling 0, and this would be its PV. Similarly, an integral ${\int }_0^2\frac{x^2\text{d}x}{x^2-1}$ can be assigned a PV of 2, by adding the integrals on (0,1–ε‎) and (1 + ε‎,2) and letting ε‎ tend to 0.