By the Hahn-Banach theorem, we know that there is a (bounded) linear functional $\phi \in (\ell_\mathbb{R}^\infty(\mathbb N))^*$ extending the usual limit in the sense that it agrees with the limit on the subspace $\ell^\infty_\text {lim}(\mathbb N) \cong \left\{(x_k)_k \subset \ell_\mathbb{R}^\infty(\mathbb N) : \exists \displaystyle\lim x_n\right\}$.
We can construct such continuous linear functionals using the Hahn-Banach theorem by defining $\operatorname{LIM}((a_n)):= L(a_1, \frac{a_1+a_2}2, \frac{a_1+a_2+a_3}3, \ldots)$ where $L$ is the extension coming from the Hahn-Banach extension theorem, and we call these Banach limits.
Now, are Banach limits the only linear extensions of $\lim$? If we require the extension to be continuous, will it force it to be a Banach limit? What about non-continuous extensions?
