How to solve the limit of the following trigonometric functions？

Question

How can i calculate limits of $\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{x \to + \infty } \cos \left[ {\left( {a + i\varepsilon } \right)x} \right] = a$, $\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{x \to - \infty } \left[ {\cos \left( {a + i\varepsilon } \right)x} \right] = 1$, $\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{x \to + \infty } \sin \left[ {\left( {a + i\varepsilon } \right)x} \right] = ia$ and $\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{x \to - \infty } \left[ {\sin \left( {a + i\varepsilon } \right)x} \right] = i$, when $a>0$ is a constant, $x$ approaches ${ \pm \infty }$ and $\varepsilon$ approaches $0$? As far as I know, $\mathop {\lim }\limits_{x \to \pm \infty } \sin ax$ and $\mathop {\lim }\limits_{x \to \pm \infty } \cos ax$ have no limit at infinity. Can someone explain the above to me? Thanks for assistance.