Is $\large{\lim_{x \to a}f(x)=L}$ The same as saying "one can get $\large{f(x)}$ as close as imaginable to $\large{L}$, by setting $\large{x}$ close enough to $\large{a}$" and vice versa "one can get $\large{x}$ as imaginable to $\large{a}$ by setting $\large{f(x)}$ close enough to $\large{L}$"?
Sorry if obvious questions, but I need to understand this before i can grasp the epsilon delta definition of a limit.
In your case, roughly speaking you could say "one can get $f(x)$ as close as imaginable to $L$, by setting $x$ close enough to $a$". About the second statement, it is not right, since there may be values of $x$ for which $f(x)=L$ but $x\neq a$, or $x$ is "very far" from $a$. For example, think of the constant function $f(x)=L$. Then, $$ f(0)=L = \lim_{x\to 50} f(x). $$
Here $a=50$, but $f(x)$ approaching $L$ does not mean $x$ approaches $50$; this is more related to injectivity of the function.