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A finite limit exists if for every ε, there exists a δ such that if

a - δ < x < a + δ

then

| f( x ) - L | < ε

It seems that an equivalent statement would be:

A finite limit exists if for every δ that defines a domain region

a - δ < x < a + δ

that the function value is limited to a range region

| f( x ) - L | < ε

Thus it could also be said that for any domain region width δ (i.e., in the "neighborhood"), then if the function value is limited to some range region L ± ε, then the finite limit exists

Would this last statement be proper?

there exists an ε such that

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# Question about the delta-epsilon definition of a limit

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