I mean why would someone deliberately create such confusing definition of limits. Why would existance of delta affects epsilon, why in proof can i put any value of delta (usually 1) and then i have to overwrite that to delta = min {1, something}. Also, proof of any question that says to prove the limit of this fx is (insert number), does not make any sense. Please explain.
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Suppose I want to convince you that the limit of sin(x)/x as x -> 0 is 1.0 (when x is in radians). We can’t simply compute sin(0)/0 because 0/0 is meaningless. Instead, we play a game. You tell me how close you want to get, and I tell you how small x has to be to get that close. Perhaps after a few iterations of the game you are convinced. Or perhaps you want to know how I pick my answers, and I can give you a formula that always works. The latter becomes a rigorous definition of the limit.
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