Delta-Epsilon Definition of Sums
Brandon H, Sahan K, Kenny S, Sally Y, Kameron W, Wiktor F
May 25, 2018
The epsilon-delta definition of a limit (or, the ε − δ definition) is a formal way of defining a
limit. Originally, the following equation: lim f (x) = L could be described by saying: As x
x→c
gets very close to c, the value of f (x) will get very close to L, but unfortunately, this kind
of definition is too vague to be used successfully in mathematical proof.
The ε − δ definition, on the other hand, defines the statement lim f (x) = L as follows:
x→c
For each ε > 0, there exists a δ > 0 such that if 0 < |x − c| < δ, then |f (x) − L| < ε.
In other words, this means that if we pick any range of y values that goes from f (x) − ε to
f (x) + ε, we can pick an x-interval x = c − δ to x = c + δ so that all values of f (x) in this
interval fall inside the range of y-values.
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