Limit Proofs Using Epsilon-Delta

1

x

6

-1+x

3

1+x

2

-4x

3

+x

5

2+x

2

1

x

+

-1+x

2

-5+6x

2

+

-2x+x

3

x

3

ϵ

1

Theorem:

lim

x∞

1

x

2

= 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

x

2

 =

1

x

2

If δ<x then

1

x

2

<

1

δ

2

Subsitute δ =

1



ϵ

and simplify

1

δ

2

= ϵ

 

1

x

2

<ϵ 

∴

lim

x∞

1

x

2

= 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.