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I have a query about the definition of limit (the ε-δone).

" Suppose f : R → R is defined on the real line and p,L ∈ R. It is said the limit of f as x approaches p is L and written

lim

_{ x→p}f(x)=L

if the following property holds:

For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − p | < δ implies | f(x) − L | < ε. "

As per the meaning of definition mentioned " given an arbitrary ε > 0,we got to find a δ > 0 small enough to guarantee that f(x) is within ε-distance from L whenever x is δ-distance of a."

Can we check for existence of a limit if we twist it a bit like this " given an arbitrary δ > 0, to find an ε> 0 small enough to guarantee that x is within δ-distance from L whenever f(x) is ε -distance of a." So here finding out an ε for a δ will allow us to say that the limit exists?

Thanks for any help..