What is $\lim\limits_{z\to\infty} \exp(z)$ with $z \in \mathbb C$?

Question

Is the exponential function even defined in the infinity point? I´ve read that it's conventionally defined equal to zero but it doesn´t make sense to me because the function is , so the limit must be the same for every path, therefore if the limit would be actually zero, when I approach for the real axis the limit would be $\infty \ne 0$, which is an absurd.