Unizor - Derivatives - Equivalence of Function Limit Definitions

Notes to a video lecture on http://www.unizor.com


Function Limit - Why Two Definitions?

Recall two definitions of function limit presented in the previous lecture.

Definition 1

Value a is a limit of functionf(x) when its argument xconverges to real number r, if for ANY sequence of argument values {x_n} converging to r the sequence of function values {f(x_n)} converges to a.

Symbolically:
∀{x_n}→r ⇒ {f(x_n}→a

Definition 2

Value a is a limit of functionf(x) when its argument xconverges to real number r, if for any positive ε there should be positive δ such that, if x is within δ-neighborhood of r(that is, |x−r| ≤ δ), then f(x)will be within ε-neighborhood of a (that is, |f(x)−a| ≤ ε).

Symbolically:
∀ ε>0 ∃ δ>0:
|x−r| ≤ δ ⇒ |f(x)−a| ≤ ε

Sometimes the last two inequalities in the above definition are specified as "less" instead of "less or equal". It makes no difference.

First of all, let's answer the question of the title of this lecture: Why two definitions?

Obviously, there were historical reasons. Mathematicians of 18th and early 19th centuries suggested different approaches to function limits and functioncontinuity that led to both definitions. Cauchy, Bolzano Weierstrass and others contributed to these definitions.
The Definition 1 seems to sound "more human", it seems more natural, though difficult to deal with if we want to prove the existence of a limit.
The Definition 2, seemingly "less human", is easier to use when proving the existence of a limit. It is more constructive.

In this lecture we will prove the equivalence of both definitions. That is, if function has a limit according to Definition 1, it is the same limit according to Definition 2 and, inversely, from existence of a limit by Definition 2 follows that this same limit complies with Definition 1.

Theorem 1
IF
for any sequence of argument values {x_n} converging to r the sequence of function values{f(x_n)} converges to a
[that is, if f(x)→a as x→raccording to Definition 1],
THEN
for any positive ε there should be positive δ such that, if x is within δ-neighborhood of r(symbolically, |x−r| ≤ δ), thenf(x) will be withinε-neighborhood of a(symbolically, |f(x)−a| ≤ ε)
[that is, it follows that f(x)→awhile x→r, according to Definition 2].

Proof

Choose any positive ε, however small, thereby fixing someε-neighborhood around limit value a.
Let's prove an existence of δsuch that, if x is closer to r thanδ, then f(x) will be closer to athan ε.
Assume the opposite, that is, no matter what δ we choose, there is some value of argument x in the δ-neighborhood of r such that f(x) is outside ofε-neighborhood of a.

Let's choose δ₁=1 and find the argument value x₁ such that|x₁−r| ≤ δ₁, while |f(x) is outside of ε-neighborhood of a.
Next choose δ₂=1/2 and find the argument value x₂ such that|x₂−r| ≤ δ₂, while |f(x) is outside of ε-neighborhood of a.
Next choose δ₃=1/3 and find the argument value x₃ such that|x₃−r| ≤ δ₃, while |f(x) is outside of ε-neighborhood of a.
etc.
Generally, on the n^th step choose δ_n=1/n and find the argument value x_n such that|x_n−r| ≤ δ_n, while |f(x) is outside of ε-neighborhood of a.

Continue this process of building sequence {x_n}.
This sequence, obviously, converges to r since|x_n−r| ≤ 1/n, but f(x_n) is always outside of ε-neighborhood of a, that is {f(x_n)} is not converging to a, which contradicts our premise that, as long as {x_n}converges to r, {f(x_n)} must converge to a.

So, our assumption that, no matter what δ we choose, there is some value of argument x in the δ-neighborhood of r such that f(x) is outside of ε-neighborhood of a, is incorrect and there exists such δ that, as soon as argument x is in theδ-neighborhood of r, value of function f(x) is withinε-neighborhood of a.
End of proof.

Theorem 2
IF
for any positive ε there is positive δ such that, if x is within δ-neighborhood of r(symbolically, |x−r| ≤ δ), thenf(x) will be withinε-neighborhood of a(symbolically, |f(x)−a| ≤ ε)
[that is, if f(x)→a as x→raccording to Definition 2],
THEN
for any sequence of argument values {x_n} converging to r the sequence of function values{f(x_n)} converges to a
[that is, it follows that f(x)→awhile x→r, according to Definition 1].

Proof

Let's consider any sequence{x_n}→r and prove that{f(x_n)}→a.
In other words, for any positiveε we will find such number Nthat for all n ≥ N the inequality|f(x_n)−a| ≤ ε is true.

Based on the premise of this theorem, there exists positive δsuch that, if |x−r| ≤ δ, it is true that |f(x)−a| ≤ ε.

Since {x_n}→r, for any δ there exist number N such that, ifn ≥ N, it is true that |x_n−r| ≤ δ.
So, for this particular N it is true that |f(x_n)−a| ≤ ε.
End of proof.