The precise definition of the limit EXPLAINED! (KristaKingMath)

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The precise definition of the limit, also called the epsilon-delta definition, is the proof of the concept of the limit. It proves the limit because it shows how, as you move closer and closer to a particular value of x, the value of the function will move closer and closer to a particular value of y.
If you want to use technical terms, what you're doing is limiting the value of delta, which is the distance you can move away from x, in order to limit the value of epsilon, which is the distance you can move away from the corresponding value along the y-axis.
To sum it up, if you found the limit some other way using calculus (like with substitution, factoring, conjugate method, L'Hospital's rule, etc.), this is the theorem that would let you prove that the value of the limit you already found is actually legitimate. It tells you why the calculus works how it does.
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