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title: CAPR01_LP10

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CAPR01_LP10
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# 1. Set and Functions

### 1.1 Set

A set is the mathematical object for a collection of different things, which can be objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.

### 1.2 Some Common sets in Math



*   $\Bbb{N}\ denote\ the\ set\ of\ all$ **Natural Numbers**,$\ \{1,2,3,\dots\}$
*   $\Bbb{Z}$, the set of all **Integers**, $\{\dots,-2,-1,0,1,2,\dots\}$
*   $\Bbb{Q}$, the set of all **Rationals**, e.g. $\{\dots,\frac{1}{2}, \frac{2}{3},\frac{23}{327}, \dots\}$
*   $\Bbb{R}$, the set of all **Real Numbers**.
*.  $\in$ called "belongs to", e.g. $1\in\Bbb{Z}$, means, 1 belongs to the set of Natual Numbers.
*   $⊆$ called "is a subset of", e.g. $\{1, 2\} ⊆ \Bbb{N}$, means, $\{1,2\}$ is a subset of the Natural Numbers.


### 1.3 Cartitian Product of Sets

For two sets A and B, we define $A \times B := \{(x, y)|x\in A\ and\ b\in B\}$


> For example, let $A = \{a, b\}$, $B = \{c, d\}$, then $A\times B = \{(a,c), (a,d), (b,c), (b,d)\}$.

**Quiz 1**: Is $(a,b) \in A\times B$?

**Quiz 2**: Find a subset of $A \times B$


### 1.4 Function

**Definition:**

> A triple $f:=(A,B,F)$ is a **Function** if:
   

1.   A and B are nonempty sets;
2.   F is a subset of $A\times B$ with the property that for each $a\in A$, there exist a **unique** $b\in B$ such that $(a,b)\in F$.

The set A is called the **Domain** of the function f, and the set B is called the co-domain of f.


**Example:**

Let $f(x):= x^2$, then $f:=(\Bbb{R}, \Bbb{R}, F)$ is a function that: $F = \{\dots, (1,1),\dots, (2,4),\dots, (3,9),\dots\} ⊆ \Bbb{R}\times \Bbb{R}$


### 1.5 Common / Classic functions in Calculas



*   **Linear** Function, $f: \Bbb{R} ↦ \Bbb{R}$, with form, $f(x) = mx + c$, where $m,c \in \Bbb{R}$ is constant, and $x\in \Bbb{R}$;
*   **Polynomial** of degree n: $p: \Bbb{R} ↦ \Bbb{R}$, with form, $p(x) = a_0 + a_1x+a_2x^2+\dots+a_nx^n$, for $x \in \Bbb{R}$;
*   **rational**, two polynomials with form $r(x)=\frac{p(x)}{q(x)}$;
*   **Power** Function: for $n \in \Bbb{N}$, $f(x) = x^n$ is called power funciton at nth power.
*   **logarithm** Function: for $x\in\Bbb{R}^+$, $f(x) = log_{a}x$, where $a\in\Bbb{R}$


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# 2. Limit and The Natural e

### 2.1 Limit of Funciton

**Definition:**

For $L\in \Bbb{R}$, we say that, **as x approaches a, the limit of f(x) is L**, if for every $ϵ > 0$, there exist a $\delta > 0$ such that 

> if $0<|x-a|<\delta$, then, $|f(x) - L| < ϵ$.

when such limit exist, we write $$\lim_{x \to a} f(x) = L$$

### 2.2 Limit at Infinity

Let $L\in \Bbb{R}$, we say that **as x approaches infinity, the limit of f(x) is L**, if for every $ϵ > 0$, there exists a number N such that, 

> if $x > N$, then $|f(x) - L| < ϵ$

when such limit exist, we write $$\lim_{x \to \infty} f(x) = L$$


**Important Example:**



*   $\lim_{x \to \infty} \frac{a}{x} = 0$, where a is constant
*   $\lim_{x \to 0} \frac{a}{x} = \infty$, where a is constant

### 2.3 **e**

**Definition:**

$$e := \lim_{n \to \infty} (1+ \frac{1}{n})^n$$


**Proposition:** $e = \lim_{t \to 0} (1 + t)^{\frac{1}{t}}$

**Proof:** let $t = \frac{1}{n}$, then $n = \frac{1}{t}$, so $n → \infty$ is equivalent to $t → 0$, then replace the variable in the definition of e, the proposition follows. $\blacksquare$

**Lemma (Compound Interest):** $e^x = \lim_{n \to \infty} (1+ \frac{x}{n})^n$

**Proof:** From definition, $e^x = (\lim_{n \to \infty} (1+ \frac{1}{n})^{n})^x = \lim_{n \to \infty} (1+ \frac{1}{n})^{nx}$, by limit laws. Then let m = nx, then $\frac{x}{m} = \frac{1}{n}$, also, $m→\infty$ as $n→∞$, so replace nx by m, and $\frac{1}{n}$ by $\frac{x}{m}$, we have $e^x = \lim_{m \to \infty} (1+ \frac{x}{m})^m$, which proved the lemma. $\blacksquare$




