If $\lim \text{sup}_{n\to\infty}X_n≠\lim \text{inf}_{n\to\infty}X_n$, can we say "limit doesn't exist"?

Question

According to the definition of the limit, if $\lim \text{sup}_{n\to\infty}X_n=A$ and $\lim \text{inf}_{n\to\infty}X_n=B$ where, $A≠B$. Terminologically, can we say "limit doesn't exist"?

Exact meaning of my question is, for example:

Let $\phi(n)$ be a Euler totient function, then we have

$$ \begin{cases} {\lim_{n\to \infty}\text{sup} \frac{\phi (n)}{n}=1} \\ {\lim_{n\to \infty}\text{inf} \frac{\phi (n)}{n}=0} \end{cases} \Longrightarrow \lim_{n\to \infty}\frac{\phi(n)}{n}{\text{doesn't exist.}}$$

Do I use this statement correctly?

Answer 1

Yes, it is correct, because whan the limit $\lim_{n\to\infty}x_n$ of a sequence $(x_n)_{n\in\mathbb N}$ exists, then$$\limsup_{n\to\infty}x_n=\liminf_{n\to\infty}x_n=\lim_{n\to\infty}x_n.$$

Answer 2

One of the definitions of $\limsup_n$ is the supremum of the partial limits, and $\liminf_n$ is the same just with infimum, if those 2 are different then there is more than one partial limit, so there is no limit

Answer 3

Yes, terminologically it is correct since the limit is a number compatible with the limit's definition. When the $\lim \text{sup}$ and $\lim \text{inf}$ are different, such a number does not exist, thus the limit does not exist.