Answer
An informal meaning of $\displaystyle \lim_{x\rightarrow a}f(x)=L$ could be formulated as
" Let the function $f$ be defined for all $x$ near $a$ except possibly at $a$.
The values of f(x) become closer and closer to L as x approaches a.
The function f may or may not be defined in x=a, and if it is, it may be some number other than L."
(sample formulation, two more in step-by-step)
Work Step by Step
Let the function $f$ be defined for all $x$ near $a$ except possibly at $a$.
If, for ANY (as small as one wants) given maximum allowed distance $\epsilon$ from the number $L$,
we can find an interval $ \lt a-\delta, a+\delta\gt $around the value $x=a$ such
all values of $f(x)$ on that interval satisfy the condition that $f(x)$ is closer to $L$ than the allowed maximum,
then we write $\displaystyle \lim_{x\rightarrow a}f(x)=L$.
A shorter formulation may be:
If, for an arbitrarily small interval around the number L,
we can find an interval with x's sufficiently close to a,
such that all f(x) are within that interval (around L)
then we write $\displaystyle \lim_{x\rightarrow a}f(x)=L$.
An informal meaning of $\displaystyle \lim_{x\rightarrow a}f(x)=L$ could be formulated as
" Let the function $f$ be defined for all $x$ near $a$ except possibly at $a$.
The values of f(x) become closer and closer to L as x approaches a.
The function f may or may not be defined in x=a, and if it is, it may be some number other than L."
(sample formulation, two more in step-by-step)
