Complex contour integration of $frac{e^{iz}}{z}$

While calculating $\int_0^\infty \frac{\sin(x)}{x} dx$ I integrated the complex function $f(z) = \frac{e^{iz}}{z}$ over the contour $C = [-R,R] \cup \gamma_R$. $\gamma_R = R e^{it}$ where $0 \leq t \leq \pi$. I had some trouble to show that $\int_{\gamma_R} f(z) \rightarrow 0$ as $R \rightarrow \infty$. Is it possible to show this without using Jordan's Lemma but instead using the ML estimate?