Limit Proofs Using Epsilon-Delta

1

x

6

x

-1+

3

x

1+x

1+

2

x

-4

3

x

+

5

x

2+

2

x

1

x

+

-1+

2

x

-5+6

2

x

+

-2x+

3

x

3

x

ϵ

1

Theorem:

lim

x∞

1

2

x

= 0

By definition,

lim

x∞

f(x) = L iff ∀ϵ>0 ∃ δ ∍ ∀x>δ, f(x)-L<ϵ

Proof:

Let ϵ>0

Choose δ =

1

ϵ

At sufficiently large values

1

2

x

=

1

2

x

If δ<x then

1

2

x

<

1

2

δ

Subsitute δ =

1

ϵ

and simplify

1

2

δ

= ϵ



1

2

x

<ϵ 

∴

lim

x∞

1

2

x

= 0

Quod erat demonstrandum.

Definitions and proofs using

ϵ

-

δ

methods are the most basic building blocks of analysis; they apply to continuity, differentiability, integrals, and series. In this Demonstration, there are several functions available and a proof is developed for each and then explained with reasoning, together with a plot that shows the concept of the proof and what is to be shown.