Epsilon-Delta Definition of Limit

​
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.

Details

Snapshot 1:
δ
is not yet small enough for the previously chosen
ϵ
Snapshot 2: discontinuous function, but one-sided limits exist
Snapshot 3: a function with no limit (with zoom); for the
ϵ
on this snapshot, no
δ
works

External Links

Limit (Wolfram MathWorld)

Permanent Citation

Ferenc Beleznay
​
​"Epsilon-Delta Definition of Limit"​
​http://demonstrations.wolfram.com/EpsilonDeltaDefinitionOfLimit/​
​Wolfram Demonstrations Project​
​Published: June 24, 2014