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I am trying to get a solid grasp of the Precise Definition of a Limit. I am having a particular hard time linking the intuition of the limit I developed a while ago to the Epsilon-Delta definition.

I understand the basics: a limit exists/is only true if and only if for every value of ε > 0 there is a δ > 0 that "encloses" a range ofxvalues whose outputs satisfy the inequality: l f (x) - L l < ε.

Now, I simply can't understand how on Earth that attests that the value of a function, f (x), approaches L as x gets infinitely close to, e.g. c ...

Here is my take on it (I hope it is at least mildly correct!):

Delta is a function of epsilon. Namely, if epsilon decreases (if we close-in on L from both sides), Delta decreases (meaning thexvalues approach c from both sides)

If the limit is true/exists, we can make epsilon as small as we want (get as close as we wish to L from both sides) thereby making Delta increasingly small (making thexvalues get closer and closer to c.) This shows that as f (x) approaches L, x approaches c from both sides: the limit is correct/true.

Am I on the right track?

Thank you very much in advance for any help whatsoever (this thing is really bothering me)!

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# Precise (Or Epsilon-Delta) Definition of a Limit

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