How to interpret limit notation $\lim\limits_{x \to a} f(x)= L$ is by most; intuitively thought of "as $x$ gets close to $a$, $f(x)$ gets close to $L$", however my lecturer said this is not correct. She told me to go away and somehow find out why, by formal definition, the intuition "$f(x)$ is close to $L$, for all $x$ sufficiently close to $a$" is correct, not the former.
I went on to find examples; Simply consider; $f(x) = x/|x|$ when $x$ tends to some number.
and to recall an emphasise; “As $x$ gets close to $a$, $f(x)$ gets close to $L$”
The emphasise on gets is important as it suggests some change towards $L$, however when investigating, as $x$ tends to some number (like $0$), $f(x) = L$, no matter where on the domain you fly. There ceases to be a case in this function where $f(x)$ moves/gets close to $L$ anywhere.
“$f(x)$ is close to $L$, for all $x$ sufficiently close to $a$” includes the idea of ‘there exists some interval’ where $f(x)$ is close to $L$.
Is that a sufficient answer to the question? I can't find anything online.

It is wise to tread carefully when semantically wording some intuition of a precise definition. 
However; in this particular case there exists at least one function that smashes the first and is consistent with the latter. That is taking the limit as $x$ tends to $0$ of;
$f(x) = xsin(1/x)$.
The statement "as x gets close to a, f(x) gets close to L" simply does not hold because no matter how close to $a$ you get, your function continues to oscillate towards and away from L. However by definition the limit does exist. 
