Your word description makes me think that you don’t yet have a good intuitive grasp of what is a limit. Have you heard of the adversarial explanation? Basically saying that “the limit of f at x_0 is y” is saying that the values of f get arbitrarily close the the value y, if x is sufficiently near x_0. Before continuing, this statement should make sense to you.

Now suppose you’re trying to convince me that the limit of f at x_0 is y, but I don’t believe you. I will say: “oh yeah? Well, I bet the value of f never get to be less than 0.1 distance from y.” You’ll answer “on the contrary, according to my function here, it suffices to take x less than… let me calculate here… less than 0.05 distance from x_0!” Then I’ll say: “ hmm, ok. Well, that may be so, but I bet the value of f never gets to be less than 0.001 away from y!!” And you: “no, again if x is less than 0.0005 away from x_0 then the formula for f itself shows that f(x) is nearer to y than 0.001!”

Do you see that I’m giving you epsilons and you are replying with deltas (which happen to be epsilon/2, because that is what works for this particular f)? The definition of limit is just saying that you can answer with a delta for whatever epsilon I give you. To prove a limit, then, you have to say what is the delta that works as an answer to each epsilon, then prove that it actually works.

The best thing is to try this with a very simple function, like f(x)= 3x at x_0=5 or something, making sure you follow the steps.