Is the notion of 'divergence to infinity in a direction' used? In $\mathbb R$ a sequence can diverge to infinity in two directions: $+\infty$ and $-\infty$. These two cases of divergence are quite different from a sequence that diverges to "nowhere", like $\{(-1)^n\}$. One can be interested, for example, in the behavior of a function in one of these two directions of infinity.
Is there a similar notion for sequences in the complex plane? (Or other spaces) Does it arise naturally in some field of study? What are possible uses for it?
I imagine, for example, that the sequence $\{ni\}$ could be said to diverge to infinity in the direction $i$, whereas the sequence $\{ne^{in\sqrt2}\}$ could be said to diverge to infinity in "every direction", or in a "divergent direction", or in a "set of directions".

Edit: Added two geometry tags, following the comment of Moishe Cohen
 A: Yes, this field of study is called geometry (metric geometry and Riemannian geometry to be more precise) and describing different ways to diverge to infinity (different directions of divergence), is quite important. Consider for instance: Gromov boundary. In their book "Encyclopedia of Distances" (pp. 116-117), M.Deza and E.Deza list 5 different ideal boundaries (and some of these 5 items have several subitems), which all capture different aspects of diverging to infinity in different directions in different spaces. (Actually, there are more boundaries than they list, once one realizes that probabilists also have ways to capture divergence to infinity, resulting in, say, Poisson, Furstenberg and Martin boundaries.) As for the complex plane, one gets the circle as the ideal boundary (in item 1 on the Deza-Deza list). This corresponds to divergence to infinity along one of the rays starting at the origin. (Each ray gives one direction.) One can start at different points in the plane, then parallel rays yield the same direction of divergence.  
