Liouville's theorem via $\int_{|z|=R}\frac{f(z)}{(z-a)(z-b)}\,dz$

Let \(f ( z )\) be bounded and analytic in C. Let \(a \neq b\) be any fixed complex numbers. Show that the following limit exists

\[ \operatorname* { l i m } _ { R \to \infty } \int _ { | z | = R } { \frac { f ( z ) } { ( z - a ) ( z - b ) } } d z . \]

Use this to show that \(f ( z )\) must be a constant (Liouville’s theorem).