Tugas TIK: Definition Of Limit

(ε, δ)-definition of limit

In calculus, the (ε, δ)-definition of limit ("epsilon-delta definition of limit") is a formalization of the notion of limit. It was first given byBernard Bolzano in 1817. Augustin-Louis Cauchy never gave an (

\varepsilon,\delta

) definition of limit in his Cours d'Analyse, but occasionally used

\varepsilon,\delta

arguments in proofs. The definitive modern statement was ultimately provided by Karl Weierstrass.

History

Isaac Newton was aware, in the context of the derivative concept, that the limit of the ratio of evanescent quantities was not itself a ratio, as when he wrote:

Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity...

Occasionally Newton explained limits in terms similar to the epsilon-delta definition. Augustin-Louis Cauchy gave a definition of limit in terms of a more primitive notion he called a variable quantity. He never gave an epsilon-delta definition of limit (Grabiner 1981). Some of Cauchy's proofs contain indications of the epsilon-delta method. Whether or not his foundational approach can be considered a harbinger of Weierstrass's is a subject of scholarly dispute. Grabiner feels that it is, while Schubring (2005) disagrees. Nakane concludes that Cauchy and Weierstrass gave the same name to different notions of limit.

Informal Statement

Let f be a function. To say that

\lim_{x \to c}f(x) = L \,

means that f(x) can be made as close as desired to L by making the independent variable x close enough, but not equal, to the value c.

How close is "close enough to c" depends on how close one wants to make f(x) to L. It also of course depends on which function f is and on which number c is. Therefore let the positive number ε (epsilon) be how close one wishes to make f(x) to L; strictly one wants the distance to be less than ε. Further, if the positive number δ is how close one will make x to c, and if the distance from x to c is less than δ (but not zero), then the distance from f(x) to L will be less than ε. Therefore δ depends on ε. The limit statement means that no matter how small ε is made, δ can be made small enough.

The letters ε and δ can be understood as "error" and "distance", and in fact Cauchy used ε as an abbreviation for "error" in some of his work. In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) to the limit point.

This definition also works for functions with more than one argument. For such functions, δ can be understood as the radius of a circle or a sphere or some higher-dimensional analogy centered at the point where the existence of a limit is being proven, in the domain of the function and, for which, every point inside maps to a function value less than εaway from the value of the function at the limit point.

Precise Statement

The

(\varepsilon, \delta)

definition of the limit of a function is as follows:

Let

f : D \rightarrow \mathbb{R}

be a function defined on a subset

D \subseteq \mathbb{R}

, let

c

be a limit point of

D

, and let

L

be a real number. Then

the function

f

has a limit

L

c

is defined to mean

for all

\varepsilon > 0

, there exists a

\delta > 0

such that for all

x

D

that satisfy

0 < | x - c | < \delta

, the inequality

|f(x) - L| < \varepsilon

holds.

Symbolically:

\lim_{x \to c} f(x) = L \iff (\forall \varepsilon > 0)(\exists \ \delta > 0) (\forall x \in D)(0 < |x - c | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon)

Example
PROBLEM 1 : Prove that $tex2html_wrap_inline157$
Begin by letting $tex2html_wrap_inline548$ be given. Find $tex2html_wrap_inline550$ so that if $tex2html_wrap_inline552$ , then $tex2html_wrap_inline554$ , i.e., $tex2html_wrap_inline556$ , i.e., $tex2html_wrap_inline558$ . But this trivial inequality is always true, no matter what value is chosen for $tex2html_wrap_inline560$ . For example, $tex2html_wrap_inline562$ will work. Thus, if $tex2html_wrap_inline552$ , then it follows that $tex2html_wrap_inline554$ . This completes the proof.

PROBLEM 2 : Prove that $tex2html_wrap_inline159$
Begin by letting $tex2html_wrap_inline548$ be given. Find $tex2html_wrap_inline550$ (which depends on $tex2html_wrap_inline574$ ) so that if $tex2html_wrap_inline610$ , then $tex2html_wrap_inline554$ . Begin with $tex2html_wrap_inline554$ and ``solve for" $tex2html_wrap_inline616$ . Then,

$tex2html_wrap_inline554$ iff $tex2html_wrap_inline620$

iff $tex2html_wrap_inline622$

iff $tex2html_wrap_inline624$

iff $tex2html_wrap_inline626$

iff $tex2html_wrap_inline628$

iff $tex2html_wrap_inline630$ .

Now choose $tex2html_wrap_inline632$ . Thus, if $tex2html_wrap_inline634$ , it follows that $tex2html_wrap_inline554$ . This completes the proof.

PROBLEM 3 : Prove that $tex2html_wrap_inline167$
Begin by letting $tex2html_wrap_inline548$ be given. Find $tex2html_wrap_inline550$ (which depends on $tex2html_wrap_inline574$ ) so that if $tex2html_wrap_inline766$ , then $tex2html_wrap_inline768$ . Begin with $tex2html_wrap_inline768$ and ``solve for" | x - 2 | . Then,

$tex2html_wrap_inline768$ iff $tex2html_wrap_inline776$

iff $tex2html_wrap_inline778$

iff $tex2html_wrap_inline780$ .

We will now ``replace" the term |3x+5| with an appropriate constant and keep the term |x-2| , since this is the term we wish to ``solve for". To do this, we will arbitrarily assume that $tex2html_wrap_inline668$ (This is a valid assumption to make since, in general, once we find a $tex2html_wrap_inline560$ that works, all smaller values of $tex2html_wrap_inline560$ also work.) . Then $tex2html_wrap_inline792$ implies that -1 < x-2 < 1 and 1 < x < 3 so that 8 < |3x+5| < 14 (Make sure that you understand this step before proceeding.). It follows that (Always make this ``replacement" between your last expression on the left and $tex2html_wrap_inline574$ . This guarantees the logic of the proof.)

$tex2html_wrap_inline802$

iff $tex2html_wrap_inline804$

iff $tex2html_wrap_inline806$ .

Now choose $tex2html_wrap_inline808$ (This guarantees that both assumptions made about $tex2html_wrap_inline560$ in the course of this proof are taken into account simultaneously.). Thus, if $tex2html_wrap_inline766$ , it follows that $tex2html_wrap_inline768$ . This completes the proof.

Source:
https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit
https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html

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Rabu, 23 Maret 2016

Definition Of Limit

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