meaning of $\lim_{x\to c}f(x)\ne L$

Question

If I want to write the negation of $\lim_{x\to c}f(x)=L$, is it considered incorrect to write $\lim_{x\to c}f(x) \ne L$ (because this indicates that the limit exists)?

Answer 1

The negation would be that for any sequence $\{x_n\}$ converging to $c$, we have that the sequence $\{f(x_n)\}$ does not converge to $L$.

If $\lim_{x \to c} f(x)$ does not exist, then the statement \begin{align*} \lim_{x \to c} f(x) \neq L \end{align*} has no meaning.