What is Limit at infinity?

Let $f$ be defined on an interval $\intco{a}{+\infty}$ and $\ell \in \R$. We say that $f$ tends to $\ell$ at $+\infty$, written $\lim\limits_{x\to+\infty} f(x) = \ell$, if every open interval containing $\ell$ contains all values $f(x)$ for $x$ large enough: for every $\varepsilon > 0$ there exists $A$ such that for all $x \geq A$, $\abs{f(x) - \ell} \leq \varepsilon$.

We say $f$ tends to $+\infty$ at $+\infty$ if for every $M \in \R$ there exists $A$ such that $f(x) \geq M$ for all $x \geq A$. Limits at $-\infty$ and limits equal to $-\infty$ are defined analogously.