The limit is that which is neither too big nor too small to be the limit.

Question

Proposed definition: $$ \lim_{x\to a} f(x) = L $$ means $L$ is the only number that is neither too big nor too small to be the limit. This can make sense only if one says precisely what "too big" and "too small" mean. Is there some published definition that does that and that is simpler than the usual $\varepsilon$-$\delta$ definition (but logically equivalent to it)?

Answer 1

The biggest number that is not too big to be the limit is one way to describe the limit superior, $\limsup_{x\to a}f(x)$, and the smallest number that is not too small to be the limit is one way to describe the limit inferior, $\liminf_{x\to a}f(x)$. Any number $l\in[\liminf,\limsup]$ is a number that is neither too big nor too small to be the limit. And if $\liminf=\limsup$, then there is only one such number, the limit $L$.

Some references that turned up on Google Books for this approach, at least for limits of sequences:

Bartle "The elements of integration and Lebesgue measure"

If the limit inferior and the limit superior are equal, then their value is called the limit of the sequence. It is clear that this agrees with the conventional definition when the sequence and the limit belong to $\mathbf R$.

Edgar "Classics on Fractals"

If the limit inferior $\underline\alpha$ and the limit superior $\overline\alpha$ coincide, then it is appropriate to speak of the convergence of the sequence and to introduce the notation $\lim A_n = \alpha$.

Sikorski "Advanced calculus: Functions of several variables"

If the limit superior and the limit inferior are equal, their common values $a$ is called the limit of $(a_n)$ and denoted by $\lim_{n\to\infty} a_n$.

Thielman "Theory of functions of real variables"

If the limit superior of a sequence $\{S_n\}$ of sets is the same set as the limit inferior of the sequence of sets, this set is called the limit of the sequence $\{S_n\}$ of sets, and it is denoted by $\lim_{n\to\infty}S_n$ or simply by $\lim S_n$.