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The Limit Laws

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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

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


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

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| $<\frac{\epsilon}{2} + <\frac{\epsilon}{2} = \epsilon$

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


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)| < $(1+|M|)\frac{ε}{2(|M|+1)}+|L|\frac{ε}{2(|L|+1)}<\frac{ε}{2}+|L|\frac{ε}{2|L|} = \frac{ε}{2}+\frac{ε}{2} = ε$∎


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  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.
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