Why does an infinite limit not exist?

According to some presentations of limits, it is proper to write "$\lim_{x\to 0}\frac{1}{x^2}=\infty$."

This does not commit one to the existence of an object called $\infty$. The sentence is just an abbreviation for "given any real number $M$, there is a real number $\delta$ (which will depend on $M$) such that $\frac{1}{x^2}\gt M$ for all $x$ such that $0\lt |x| \lt \delta$." It turns out that we often wish to write sentences of this type, because they have important geometric content. So having an abbreviation is undeniably useful.

On the other hand, some presentations of limits forbid writing "$\lim_{x\to 0}\frac{1}{x^2}=\infty$." Matter of taste, pedagogical choice. The main reason for choosing to forbid is that careless manipulation of the symbol $\infty$ all too often leads to wrong answers.

A limit

$$\lim_{x\to a} f(x)$$

exists if and only if it is equal to a number. Note that $\infty$ is not a number. For example $\lim_{x\to 0} \frac{1}{x^{2}} = \infty$ so it doesn't exist.

When a function approaches infinity, the limit technically doesn't exist by the proper definition, that demands it work out to be a number. We merely extend our notation in this particular instance. The point is that the limit may not be a number, but it is somewhat well behaved and asymptotes are usually worth note.