WOLFRAM|DEMONSTRATIONS PROJECT

Epsilon-Delta Definition of Limit

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Choose a function f
so that the limit exists
at x=a.
Looking for a
two-sided limit
zoom
Definition:
L
The number L is the limit of f
as x approaches a,
if and only if for all ϵ > 0,
ϵ
there is a δ > 0
δ
such that if 0 < |x - a| < δ,
x
then |f(x) - L| < ϵ.
Did you succeed in finding an
appropriate δ > 0 for all ϵ > 0
for your choosen L?
One of the key concepts of calculus is the limit of a function. Informally, a function
f:
has a limit
L
at a point
a
if the value
f(x)
gets close to a fixed number
L
as
x
gets close to
a
. This Demonstration illustrates a more formal definition of limit, usually referred to as the
ϵ
-
δ
definition. The arrangement of the sliders highlights the importance of the wording of the definition.