Using the definition in Bartle's Introduction to Real Analysis, I am trying to gain an intuitive understanding of limits that tend to infinity.
Given Definition:
Let ($x_n$) be a sequence of real numbers.
(i) We say that ($x_n$) tends to $\infty$, and write $lim(x_n) = +\infty$ , if for every $\alpha \in \Bbb R$ there exists a natural number $K(\alpha)$ such that if $n \ge K(\alpha)$, then $x_n > \alpha$.
(ii) We say that $(x_n)$ tends to $- \infty$, and write $lim(x_n) = - \infty$, if for every $\beta \in \Bbb R $ there exists a natural number $K(\beta)$ such that if $n \ge K(\alpha)$, then $x_n < \beta$.
My question is this: Can anyone put this into words or paraphrase their understanding of this concept? I'm having trouble grasping the concept of these kinds of limits with the formal definitions. Apologies if this is more of a 'soft' question.
As an example question, I know that the lim$(\sqrt{n^2+2} = \infty$ however I don't understand why this is so.