I heard that the reason epsilon delta method is used is that the original limit is intuitive.
Then why is intuition wrong? Is there any way of proving that? Is there any example which is solved not by limits, but only by the epsilon-delta method?
2026-03-30 12:19:32.1774873172
Why is the epsilon-delta method needed?
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Intuition fails to get the right answers often (Cantor's theorem and the Weierstrass's function are examples) so mathematics needs to be made rigorous to avoid these, precise definitions like the epsilon-delta definition of a limit need to be used.
I think it is quite intuitive, it states that a function $f$ reaches a limit $L$ at a point $a$ if and only if we can make $f$ as close to $L$ as we want it to be (within epsilon. That's why it's for all epsilon, any can be chosen for the desired precision) if we consider points which are close enough to $a$ (this is the delta part. There is (there exists) some delta which makes sure $f(x)$ is within epsilon of $L$ if $x$ is within delta of $a$, we have to make $x$ close enough to $a$).