Epsilon-delta definition of limits, states that the limit of f(x) at x=c equals L if, for any ε>0, there's a δ>0 ensuring that when the distance between x and c is less than δ, the distance between f(x) and L is less than ε.
Now, here the first part of statement says that δ is a function of ε, hence ε should be the independent variable while δ is dependent variable which is dependent on ε. Now, coming to the second part shouldn't it say that whenever distance between f(x) and L is less than ε,then the distance between x and c is less than δ and not the converse as defined in the statement?? As it is ε which is independent and not δ hence we will not know δ value before we know ε value as δ is a function of ε.
or conversely, we can just initially define ε itself as a function of δ instead of the other way around.
Let's look at the implication statement you are asking about:
You seem to think that there is something about this implication that tells us that $\epsilon$ ought to depend on $\delta$. But that is not the case.
In this statement, there some constants that are given to us ahead of time (the function symbol $f$, the $c$, and the $L$), and there are three variables $x$, $L$ and $\epsilon$.
There is nothing in this statement to tell us which of the three variables ought to be independent, or dependent, or anything else like that. They are free variables. Assuming that $f$ and $c$ and $L$ have been given, we can freely plug in any values of $x$, $\delta$, $\epsilon$ that we like (restricting $x$ to the domain of $f$) and use them to test the truth of this implication.
For instance suppose that $f(x)=x^2$, $c=2$, and $L=4$.
The implication statement is true for $x=2.1$, $\delta=.5$, and $\epsilon = 10000$, because both the hypothesis $d(x,c)<\delta$ and the conclusion $d(f(x),L)<\epsilon$ are true).
On the other hand the implication statement is false for $x=2.1$, $\delta=.5$, and $\epsilon = .0001$, because the hypothesis $d(x,c)<\delta$ is true whereas the conclusion $d(f(x),L)<\epsilon$ is false.
So, sometimes this implication is true, and sometimes it is false.
The mathematical meaning of a limit is only determined when we put the quantifiers in front:
In my second example where $\epsilon=.0001$, I would trace the falsity of the implication to a bad choice of $\delta$. The definition of limits tells me that there is a $\delta$ that depends on $\epsilon = .0001$, it just so happens that $\delta = .5$ is not it. I would have done much better if I had followed the proof scheme for limit proofs: first use the given values of $x$ and $\epsilon$ to find an appropriate value of $\delta$, hence the value of $\delta$ you find will depend on the $\epsilon$; and second use the value of $\delta$ you found (and the given values of $x$ and $\epsilon$) to prove that the implication is true.