$\displaystyle{\lim_{z \to ia} \cos z}$ and $\displaystyle{\lim_{z \to ia} \Re (e^{iz})}$, are they equal or not?
Solution 1:
The problem is not the limit, but that the identity $\cos z = \Re(e^{iz})$ holds only for real numbers $z$: $$ \Re(e^{iz}) = \frac 12 \left( e^{iz} + e^{-i \bar z}\right) \\ \cos(z) = \frac 12 \left( e^{iz} + e^{-i z}\right) \\ $$ are equal if and only if $$ e^{-i \bar z} = e^{-i z} \iff z = \bar z \iff z \in \Bbb R \, . $$