That's not even how it should be written with quantifiers. Check your definitions.
Where you have your first ⇒ sign, you need to replace that with ∀x .

p ⇒ q can be thought of as "if p then q".

So, I'd say:

For any positive value of epsilon, there exists a positive number delta, such that for any x, IF the distance of x from c is less than delta (and greater than zero) THEN the distance of f(x) from L is less than epsilon.
the word value before variables is usually omitted from my experience, with L usually being mentioned as the limit in this case. Also it may be stylistic, but in my region we use minus instead of subtract.
by
2 errors:

(1) "x subtract c" should be "the difference between x and c", because you're taking the absolute value.

(2) You've stated the implication incorrectly. It should be:

"...such that _if_ the difference between x and c less than delta, _then_ the difference between f(x) and L is less than epsilon."