I know when you wanna prove that the limit of some function like $f(x)$ at $a$ exists you must prove a proposition of the following form:
$$(\forall \varepsilon >0)(\exists \ \delta >0)(0<|x-a|<\delta \ \Rightarrow \ |f(x)-l|<\varepsilon )$$
But what if you wanna prove that the limit of $f$ at $a$ does not exist. Do I need to negate the above expression and then reach it by recursive reasoning? Then what really is the negation of this expression? Should it lead to non-uniqueness of $l$?
Moreover I know that "Infinite limits" -which are limits with a value of infinity- are also classified as non-existent limits. Then how could we prove they don't exist using $\varepsilon$ and $\delta$?
Thank you in advance.