Why does an infinite limit not exist? I read in Stewart "single variable calculus" page 83 that the limit $$\lim_{x\to 0}{1/x^2}$$ does not exist. How precise is this statement knowing that this limit is $\infty$?. I thought saying the limit does not exist is not true where limits are $\infty$. But it is said when a function does not have a limit at all like  $$\lim_{x\to \infty}{\cos x}$$.
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.
Ok ...
I think there is a difference between saying the limit doesn't exsist and equal to infinity 
Limit f(x) as x approaches to x' (exsist ) iff lim f(x ) when x approaches to x' from the right = limit f(x)  as x reproaches to x' from the left
And   in the above limit in your question  is  exist because  limit from the right  equal the limit from the left equal to infinity +
  but if you take f(x)= 1\x  in this  case the limit doesn't exsist  the limit from the right =infinity+  but the limit from the left equal to infinity - 
Please don't hesitate to discuss this with me.