Cauchy's Definition of a Limit

This is the most feared and dreaded by most beginning calculus students

It is, however, the most important definitions in Calculus.

It defines a limit of a function in precise mathematical terms.
In this form it is used to prove many limit statements (including most limit theorems).

Also used to define other concepts which have limits for definitions

Unfortunately, this is a big "stubbling block" for most beginning calculus students. Even experienced students have much difficulty using it.

In informal words, to justify a limit is L as x approaches a fixed number a, one must demonstrate that no matter what neighborhood size Î (epsilon) of L you choose, you will always be able to find a neighborhood size d (delta) of a, such that, all x's in that d-neighborhood of a, will give functional values within the Î (epsilon) distance of L (are mapped into that Î-neighborhood of L)

Note: Informally, think of yourself as working at delta corp ($1,000/day). Your job is to respond to callers that send in different size epsilons (neighborhoods of L, the limit of the day). You need to respond to each caller by giving them a delta (how close to a, the value x is approaching) that will satisfy each epsilon. If the phone does not ring, you have nothing to do. Doing you homework & knowing the limit of the day, you can figure out what response will work in advance. This way, your job is made quite easy. Figuring this correct response requires you to prove the limit of the day (which I will show in this discussion)

We do not want x to equal a, so the definition eliminates x=a by the statement 0<|x-a|. So, we are dealing with deleted neighborhoods of a

a limit, alone, does not depend upon what happens at the point it approaches only close to it on both sides.(for basic limits)

however, some concepts that are defined in terms of limits will. (i.e., continuity) (we need the limit to equal the functional value there)

Here is the full statement of Cauchy's definition:

To show (or prove) that lim f(x)= L as x → a, one must show the following:

"(For all)Î>0, there exists ($) d>0, such that ('), if 0< |x - a| < d, then
then |f(x) - L| <Î

Special symbols, included, are used for "for all" (universal quantifier), "there exists"(existential quantifier), and "such that". Also, an if - then statement can be symbolized with a right arrow.

Example of a simple proof:

Prove: lim (3x-5) = 7 as x→4 using Cauchy's definition.

Proof: To Show: "Î>0, $d>0, ', if 0< |x - 4| <d, then |(3x-5) - 7| <Î.

Analysis:

Let Î>0 be given.(phone rings). Consider |(3x-5) - 7| = |3x-12| = |3(x-4)| = 3 |x-4|

By forcing 3 |x-4| <Î, we see that |x-4| <Î/3

Claim: So, choose d ≤ Î/3.

Synthesis: if |x-4| < Î/3, then 3|x-4| <Î, or |(3x-5) -7| <Î.

Which satisfies the conditions of Cauchy's definition

~Note: So, at Delta corp, just divide the incoming call number by 3 & all callers will be satisfied & you should get a raise in pay. Also note that your response is the largest delta that will work. If this neighborhood size gets you where you want, then any smaller one would also.

Most teachers do not require the synthesis when testing students

This is a simple proof. Many proofs are not and require other conditions