Precise definition of a limit- do i understand this concept? Can someone tell me if I'm thinking about this definition correctly?  I tried to summarize the precise definition of a limit by saying:
$F(x)=L$ over some range of $y$ values correlates to the range of possible $x$ values that make a limit statement plausible.  
So as the range of $f(x)$ values get smaller and closer to $L$ , the range of $x$ values also get smaller and closer to $a$.  And the same situation if one range increases, the other range also increases.
Even if I am on the right track, can someone explain it in different way so that I may better understand this?


So as the range of f(x) values get smaller and closer to L , the range of x values also get smaller and closer to a

No offense, but that is a pretty confused sentence, if you ask me.
I don't understand what exactly you mean by most of the words you use, so I will rather write my own explanation.

The definition of a limit is that 

$\lim_{x\to a} f(x)=L$ is true if and only if, for every $\epsilon>0$, there exists $\delta>0$ such that if $0<|x-a|<\delta$, then $|f(x)-L|<\epsilon$.

This definition tells you that, no matter what $\epsilon$ I pick, you can find me a $\delta$ such that $f(x)$ will be $\epsilon$ - close to $L$ if $x$ is $\delta$-close to $a$.
With fewere variables, this would translate to

I can get $f(x)$ to be as close to $L$ as I want, if I only put $x$ close enough to $a$.

