JustToThePoint English Website Version
JustToThePoint en español
JustToThePoint in Thai

The Limit Laws

If you torture data long enough, it will confess to anything or, in other words, there are different types of lies: white lies, seemingly small exaggerations and half-truths, damn lies, out-of-context information, and misleading statistics, #Anawim, justtothepoint.com.

Recall

Definition. A function f is a rule, relationship, or correspondence that assigns to each element of one set (x ∈ D), called the domain, exactly one element of a second set, called the range (y ∈ E).

The pair (x, y) is denoted as y = f(x). Typically, the sets D and E will be both the set of real numbers, ℝ. A mathematical function is like a black box that takes certain input values and generates corresponding output values (Figure E).

 

Very loosing speaking, a limit is the value to which a function grows close as the input get closer and closer to some other given value.

One would say that the limit of f, as x approaches a, is L, $\lim_{x \to a} f(x)=L$. Formally, for every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − a | < δ implies that | f(x) − L | < ε. In other words, f(x) gets closer and closer to L, f(x)∈ (L-ε, L+ε), as x moves closer and closer -approaching closer but never touching- to a (x ∈ (a-δ, a+δ), x≠a)) -Fig 1.a.-

Definition. Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a, then we say that, $\lim_{x \to a} f(x) = L$ if

$\forall \epsilon>0, \exists \delta>0: 0<|x-a|<\delta, implies~ |f(x)-L|<\epsilon$

Or

$\forall \epsilon>0, \exists \delta>0: |f(x)-L|<\epsilon, whenever~ 0<|x-a|<\delta$    

The Limit laws

Let f(x), g(x) be functions defined on an interval that contains x = a, except possibly at x = a, assume that L and M are real numbers such that $ \lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$. Let c be a constant. Then, each of the following statements holds:

Proof: Let $\epsilon>0$

$\exists \delta_1>0: 0<|x-a|<\delta_1, implies~ |f(x)-L|<\frac{\epsilon}{2}$

$\exists \delta_2>0: 0<|x-a|<\delta_2, implies~ |g(x)-M|<\frac{\epsilon}{2}$

Let’s choose $\delta = min (\delta_1, \delta_2).$

$\forall \epsilon>0, \exists \delta>0: 0<|x-a|<\delta$ ⇒[By the triangle inequality, |a+b|≤|a|+|b|] |f(x)+g(x)-L-M| ≤ |f(x)-L|+ |g(x)-M| $<\frac{\epsilon}{2} + <\frac{\epsilon}{2} = \epsilon$

$\lim_{x \to 2} x^2·(x+1) = \lim_{x \to 2} x^2·\lim_{x \to 2} x+1 = 4·3 = 12.$
$\lim_{x \to 2} (6x-x^3)·(x^2+3x-1) = \lim_{x \to 2} 6x-x^3·\lim_{x \to 2} x^2+3x-1 = 4·9 = 36.$ $\lim_{x \to 5} (2x^2-3x+4) = \lim_{x \to 5}(2x^2) - \lim_{x \to 5}(3x) + \lim_{x \to 5}(4) = 2·\lim_{x \to 5}(x^2) - 3·\lim_{x \to 5}(x) + \lim_{x \to 5}(4) = 2·5^2-3·5+4 = 39.$

Proof.

Let $\epsilon>0$

$\exists \delta>0: 0<|x-a|<\delta, implies~ |f(x)g(x)-LM|<\epsilon$

|f(x)g(x)-LM| = |f(x)g(x) -Lg(x) + Lg(x) -LM| = |g(x)(f(x)-L) + L(g(x)-M)| ≤[Triangle inequality] |g(x)(f(x)-L)| + |L(g(x)-M)| = |g(x)(f(x)-L)| + |L(g(x)-M)| = |g(x)||(f(x)-L)| + |L||(g(x)-M)|

This is the 🔑 tricky part:

  1. $lim_{x \to a} g(x)$ = M ⇒ Let $\frac{ε}{2(|L|+1)}, \exists \delta_1>0: 0<|x-a|<\delta_1, implies~ |g(x)-M|<\frac{ε}{2(|L|+1)}.$
  2. Let $ε = 1> 0, \exists \delta_2>0: 0<|x-a|<\delta_2, implies~ |g(x)-M|<1⇒|g(x)| = |g(x) -M +M| ≤ |g(x) -M| + |M|< 1 + |M|.$
  3. $lim_{x \to a} f(x)$ = L ⇒ Let $\frac{ε}{2(|M|+1)}, \exists \delta_3>0: 0<|x-a|<\delta_3, implies~ |f(x)-L|<\frac{ε}{2(|M|+1)}.$

We choose δ = min{δ1, δ2, δ3}

|f(x)g(x)-LM| ≤ |g(x)||(f(x)-L)| + |L||(g(x)-M)| < $(1+|M|)\frac{ε}{2(|M|+1)}+|L|\frac{ε}{2(|L|+1)}<\frac{ε}{2}+|L|\frac{ε}{2|L|} = \frac{ε}{2}+\frac{ε}{2} = ε$∎

Bibliography

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
  1. NPTEL-NOC IITM, Introduction to Galois Theory.
  2. Algebra, Second Edition, by Michael Artin.
  3. LibreTexts, Calculus.
  4. Field and Galois Theory, by Patrick Morandi. Springer.
  5. Michael Penn, and MathMajor.
  6. Contemporary Abstract Algebra, Joseph, A. Gallian.
  7. YouTube’s Andrew Misseldine: Calculus, College Algebra and Abstract Algebra.
  8. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.
  9. blackpenredpen.
Bitcoin donation

JustToThePoint Copyright © 2011 - 2024 Anawim. ALL RIGHTS RESERVED. Bilingual e-books, articles, and videos to help your child and your entire family succeed, develop a healthy lifestyle, and have a lot of fun.

This website uses cookies to improve your navigation experience.
By continuing, you are consenting to our use of cookies, in accordance with our Cookies Policy and Website Terms and Conditions of use.