Epsilon-Delta Definition of a Limit
Author: Jason McCullough

For a given function f(x), in order to say the limit of f(x) as x approaches a is L, we need to know that for Every ε > 0, there exists a corresponding δ > 0, such that if |x - a| < δ, then |f(x) - L| < ε.

Use the sliders to change the value of δ to find a value that works with ε=2.  Then try shrinking ε and finding a corresponding δ for the new ε.

You can drag the point a around to see behavior at other points.
Hold CNTR and use the mouse and school wheel to change the displayed window.
Initially, f(x) = 1/8 x^3.
Type in "f(x) = " to try a different function. 
(Note: This demo assumes f(x) is continuous and monotone in the interval (x - δ, x + δ).  )

a

a

L

a - δ

a + δ

L + ε

L - ε

x + δ

2δ

2ε

   Ce  ~ δ  ~garantit  ~que  ~si \mid x - a \mid < δ, alors \mid f(x) - L \mid < ε

 Ce δ ne suffit pas (un plut petit δ pourrait aller)

Source : http://users.rider.edu/~jmccullough/

f(x) = (1 / 8 * x^(3))
A = Point(xAxis)
T = Point(xAxis)
δ = 0.5 (slider from 0.01 to 1, step 0.01)
B: (x(T), f(x(T)))
C: (0, y(B))
a: x = x(T) - δ
b: x = x(T) + δ
ε = 1.81 (slider from 0.01 to 2, step 0.01)
c: y = y(C) + ε
d: y = y(C) - ε
h: abs(f(x(T) + δ) - f(x(T))) < ε
e: abs(f(x(T) - δ) - f(x(T))) < ε
goodDelta = If(e ∧ h, 1, 0)
D: (x(T) + δ, 1000)
E: (x(T) + δ, -1000)
F: (x(T) - δ, 1000)
G: (x(T) - δ, -1000)
H: (1000, y(C) + ε)
I: (-1000, y(C) + ε)
J: (1000, y(C) - ε)
K: (-1000, y(C) - ε)
poly2, h_1, i, k, j = Polygon(H, I, K, J)
poly1, d_1, e_1, g_1, f_1 = Polygon(D, E, G, F)
L = Intersect(c, yAxis)
M = Intersect(d, yAxis)
N = Intersect(xAxis, a)
O = Intersect(b, xAxis)
P: (x(T) - δ, -1)
Q: (x(T) + δ, -1)
l = Segment(P, Q)
R: (-4, y(B) + ε)
S: (-4, y(B) - ε)
m = Segment(R, S)
g = Function(f, x(T) - δ, x(T) + δ)