want to prove that if $f(x)$ is a real rational function whose denominator is different from zero for all real x and is of degree at least two units higher than the degree of the numerator. Then the limits in
$$\int\limits_{ - \infty }^\infty {f(x)dx = \mathop {\lim }\limits_{a \to - \infty } } \int\limits_a^0 {f(x)dx} + \mathop {\lim }\limits_{b \to \infty } \int\limits_0^b {f(x)dx} $$
exist. In addition, if $S$ is the semicircular arc shown in the figure below
we have on $S$ $$\left| {f(z)} \right| < {k \over {{{\left| z \right|}^2}}}$$ if $$\left| z \right| = R > {R_0}$$ for sufficiently large constants $k$ and ${R_0}$