Although it is a more philosophical phenomenon rather than mathematical one, the question of the infinity in the study of limit is interesting for me. If there is no limit of the infinity why we define the fact that limit in the infinity can exist? For example:
$ \lim_{x \rightarrow 0 } \frac{1}{|x|} = \infty $
As can be seen in the example above, the limit exists even we get the answer as infinity.
The following
$$\lim_{x \rightarrow 0 } \frac{1}{|x|} = \infty$$
is just a symbolic way to say that, assuming $x$ values sufficiently near to $0$, we can make the expression $\frac{1}{|x|} $ arbitrarly larger than every assigned quantity $M \in \mathbb{R}$.