Definition:Neighborhood of Infinity
Topology
Let $X$ be a non-empty topological space.
A neighborhood of infinity is a subset of $X$ which contains the complement of a closed and compact subset of $X$.
Real Analysis
Neighborhood of Positive Infinity
A neighborhood of $+\infty$ is a subset of the set of real numbers $\R$ which contains an interval $\openint a \to$ for some $a \in \R$.
That is, a subset which contains all sufficiently large real numbers.
Neighborhood of Negative Infinity
A neighborhood of $-\infty$ is a subset of the set of real numbers $\R$ wich contains an interval $\openint \gets a$ for some $a \in \R$.
That is, a subset which contains all sufficiently large negative real numbers.
Complex Analysis
A neighborhood of $\infty$ in $\C$ is a subset of the set of complex numbers $\C$ wich contains a set of the form $\{z \in \C : |z| > r\}$ for some $r\in\R$.
That is, a subset which contains all complex numbers whose modulus is sufficiently large.