Residues — a vanishing contour integral around a simple pole

Let $S$ be a finite subset of $\mathbb{C}$ containing 0. Let $f:\mathbb{C}\setminus S\to\mathbb{C}$ be a holomorphic function which has a simple pole at 0. For $R>0$, let $\gamma_R$ denote the path $\gamma_R(t)=Re^{2\pi it}$ for $t\in[0,1]$. Which of the following statements is necessarily true?