Answer
The statement is not sufficient to define the limit of $f(x)$ as $x$ approaches $c$. See the counter example below.
Work Step by Step
"The number $L$ is the limit of $f(x)$ as $x$ approaches $c$ if $f(x)$ gets closer to $L$ as $x$ approaches $c$."
This statement is not sufficient to define the limit of $f(x)$ as $x$ approaches $c$. Take this example:
$$f(x)= x^4-9$$
As $x$ approaches $0$, $f(x)$ actually does not just come closer to $-9$. It also comes closer to $-10, -11, -12, -13$ and so so on. Yet we know that $\lim_{x\to0}(x^4-9)=-9$, and there is only one value to that.
Saying $f(x)$ gets closer to $L$ as $x$ approaches $c$, hence, is not enough to find $\lim_{x\to c}f(x)$, as in the case of $\lim_{x\to0}(x^4-9)$ above, there are too many $L's$ that $f(x)$ "gets closer to" as $x$ approaches $c$.
