# Week 2-3 - Functions & Limits

## 1. Functions

**Deﬁnition**: A function`f`

is a rule that assigns to each element`x`

in a set`D`

exactly**one**element, called $f(x)$, in a set`E`

.- The set
`D`

is called the**domain**of the function. - The number $f(x)$ is the value of
`f`

at`x`

and is read "f of x". - The
**range**of`f`

is the set of all possible values of $f(x)$ as`x`

varies throughout the domain. - A symbol that represents an arbitrary number in the
*domain*of a function`f`

is called an**independent variable**. A symbol that represents a number in the*range*of`f`

is called a**dependent variable**.

- The set
Four ways to represent a functions

- verbally (by a description in words)
- numerically (by a table of values)
- visually (by a graph)
- algebraically (by an explicit formula)

A catalog of essential functions

**linear function**- $f(x) = mx+b$

**polynomials**- $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{2}x^{2}+a_{1}x^{1}+a_{0}$
- The domain of any polynomial is $\mathbb{R} = (-\infty, + \infty)$.
- If the leading coefficient $a_{n}\ne 0$, then the degree of the polynomial is
**n**.`n=2`

: quadratic function`n=3`

: cubic function

**power function**- $f(x) = x^a$
`a = n`

, where`n`

is a positive integer`a = 1/n`

, where`n`

is a positive integer. It's a**root function.**`a = -1`

:**reciprocal function**

- $f(x) = x^a$
**rational function**- A rational function
`f`

is a ratio of two polynomials:- $f(x)=\frac{P(x)}{Q(x)}$, $\{Q(x)\ne0\}$
- $f(x)=\frac{2x^{4}-x^{2}+1}{x^{2}-4}$, $\{x|x \ne \pm 2\}$:

- A rational function
**algebraic function**- A function
`f`

is called an**algebraic function**if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots)

- A function
**trigonometric function**- $f(x)=\sin{x}$

**exponential function**- $f(x)=b^x$
- law of exponential function:
- $b^{x+y}=b^x+b^y$
- $b^{x-y}=\frac{b^x}{b^y}$
- $(b^{x})^{y}=b^{xy}$
- ${ab}^{x}=a^{x}b^{x}$

**logarithmic function**- $f(x)=\log_{b}{x}$

### 1.1. Combinations of Functions

- $(f+g)(x)=f(x)+g(x)$
- $(f-g)(x)=f(x)-g(x)$
- $(fg)(x)=f(x)g(x)$, the domain of
`fg`

is $A \cap B$ - $(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$, the domain of
`f/g`

is $\{x \in A \cap B\ |\ g(x) \ne 0\}$. - $(f \circ g)(x) = f(g(x))$
- composition (or composite) of
`f`

and`g`

, denoted by $f \circ g$ (“f circle g”).

- composition (or composite) of

### 1.2. Inverse of Functions

- Definition:
- If a function maps every input to exactly
**one**output, an inverse of that function maps every “output” to exactly**one**“input.”

- If a function maps every input to exactly
- denoted by$f^{-1}$ , and read “
**f inverse**”. - to function $N=f(t)$, the inverse function will be $t=f^{-1}(N)$.

#### one-to-one functions

- A function is
**one-to-one**if for every value in the**range**(`f(x)`

), there is exactly one value in the**domain**(`x`

).- domain of $f^{-1}$ = range of $f$
- range of $f^{-1}$ = domain of $f$
- for example: $f(x)=x^3$ is a one-to-one function, $f(x)=x^2$ is not.

## 2. Limits

- Definition:
- $\displaystyle\lim_{x \to a}{f(x)}=L$
- the limit of
`f(x)`

, as`x`

approaches`a`

, equals`L`

- if we can make the values of
`f(x)`

arbitrarily close to`L`

(as close to`L`

as we like) by restricting`x`

to be sufficiently close to`a`

(on either side of`a`

) but**NOT equal**to`a`

.(This means that in finding the limit of`f(x)`

as`x`

approaches`a`

, we never consider`x = a`

.)

### 2.1. Limit Laws

Suppose that

`c`

is a constant and the limits $\displaystyle\lim_{x \to a}{f(x)}$ and $\displaystyle\lim_{x \to a}{g(x)}$ exist, Then:- $\displaystyle\lim_{x \to a}{[f(x)+g(x)]}=\lim_{x \to a}{f(x)}+\lim_{x \to a}{g(x)}$
- $\displaystyle\lim_{x \to a}{[f(x)-g(x)]}=\lim_{x \to a}{f(x)}-\lim_{x \to a}{g(x)}$
- $\displaystyle\lim_{x \to a}{cf(x)}=c\lim_{x \to a}{f(x)}$
- $\displaystyle\lim_{x \to a}{[f(x)g(x)]}=\lim_{x \to a}{f(x)}\cdot \lim_{x \to a}{g(x)}$
- $\displaystyle\lim_{x \to a}{\frac{f(x)}{g(x)}}=\frac{\displaystyle\lim_{x \to a}{f(x)}}{\displaystyle\lim_{x \to a}{g(x)}}$, if $\displaystyle\lim_{x \to a}{g(x)} \ne 0$

These five laws can be stated verbally as follows:

**Sum Law**: The limit of a sum is the sum of the limits.**Difference Law**: The limit of a difference is the difference of the limits.**Constant Multiple Law**: The limit of a constant times a function is the constant times the limit of the function.**Product Law**: The limit of a product is the product of the limits.**Quotient Law**: The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0).

### 2.2. Squeeze Theorem

- if $g(x) \le f(x) \le h(x)$ and $\displaystyle\lim_{x\to{a}}g(x) = \lim_{x\to{a}}h(x) = L$, then $\displaystyle\lim_{x\to{a}}f(x)=L$
- Sample: to prove $\displaystyle\lim_{x\to{0}}\frac{\sin(x)}{x}=1$
- we know:
- $\displaystyle\lim_{x\to{0}}\cos(x)=1=\lim_{x\to{0}}1$
- $\cos(x) \le \frac{\sin(x)}{x} \le 1$

- then:
- $\displaystyle\lim_{x\to{0}}\cos(x) \le \lim_{x\to{0}}\frac{\sin(x)}{x} \le 1$

- so:
- $\displaystyle\lim_{x\to{0}}\frac{\sin(x)}{x}=1$

- we know:

### 2.3. Continuity

- Definition 1:
- A function
`f`

is**continuous at a number a**, if $\displaystyle\lim_{x \to a}f(x)=f(a)$- Notice that this
**Definition**implicitly requires three things if`f`

is continuous at`a`

:`f(a)`

is defined (that is,`a`

is in the domain of`f`

)- $\displaystyle\lim_{x \to a}f(x)$ exists
- $\displaystyle\lim_{x \to a}f(x) = f(a)$

- Notice that this

- A function
- Definition 2:
- A function
`f`

is**continuous from the right at a number a**if $\displaystyle\lim_{x \to a^{+}}f(x) = f(a)$, and`f`

is**continuous from the left at a**if $\displaystyle\lim_{x \to a^{-}}f(x) = f(a)$

- A function
- Definition 3:
- A function
`f`

is**continuous on the interval**`(a, b)`

, if for all points`c`

so that`a < c < b`

,`f(x)`

is continuous at`c`

.**close intervals**:- To say "
`f(x)`

is**continuous on the interval**`[a, b]`

", means:`f(x)`

is**continuous on the interval**`(a, b)`

- $\displaystyle\lim_{x \to a^{+}}f(x) = f(a)$
- $\displaystyle\lim_{x \to b^{-}}f(x) = f(b)$

- To say "

- A function

#### The Intermediate Value Theorem

- Suppose that
`f`

is continuous on the closed interval`[a, b]`

and let`N`

be any number between`f(a)`

and`f(b)`

, where $f(a) \ne f(b)$. Then there exists a number`c`

in`[a, b]`

such that $f(c) = N$. - For example: how to approximate root two?
- use function $f(x)=x^2-2$
- when $x=\sqrt{2}$, this function equals
`0`

, - so i just need to look for a positive value, that I can plug into this function to make it equals zero.
- We know that
`f(1)=-1<0`

and`f(2)=2>0`

. So base on**The Intermediate Value Theorem**, there must be a value in domain (1, 2) that exist`c`

such that`f(c) = 0`

. - to continue calculating
`f(1.5)=0.25>0`

, we got $c \in (1, 1.5)$, then $c \in (1.4, 1.5)$ ... then we are getting closer and closer to $\sqrt{2}$.

- when $x=\sqrt{2}$, this function equals

- use function $f(x)=x^2-2$

#### Fixed Point

- Definition
- A
**fixed point**of a function`f`

is a number`c`

in its domain such that`f(c) = c`

. (The function doesn’t move`c`

; it stays fixed.)- if
`f(x)`

continuous on`[0,1]`

, and $0 \le f(x) \le 1$, then there is an`x`

in domain`[0, 1]`

, exist`f(x) = x`

.- To Prove:
- Assumption:
`g(x) = f(x) - x`

, so`g(x)`

is continuous `g(0) = f(0) - 0 >= 0`

`g(1) = f(1) - 1 <= 1`

- Base on the IVT(Intermediate Value Theorem), there must be an
`x`

such that`g(x) = 0`

, which is`f(x) = x`

.

- if

- A

### 2.4. Infinity

Definition

- $\displaystyle\lim_{x \to a}f(x) = \infty$ means that
`f(x)`

is as large as you like provides`x`

is close enough to`a`

.- $\displaystyle\lim_{x \to \pi/2}\tan(x) = \infty$

- $\displaystyle\lim_{x \to \infty}f(x) = L$ means that
`f(x)`

is close enough to`L`

provided`x`

is large enough.- $\displaystyle\lim_{x \to \infty}\tan^{-1}(x) = \frac{\pi}{2}$

- if $\displaystyle\lim_{x \to \infty}f(x) = L$ or $\displaystyle\lim_{x \to -\infty}f(x) = L$, then the line
`y = L`

is called a**horizontal asymptote**of the curve`y = f(x)`

:

- $\displaystyle\lim_{x \to a}f(x) = \infty$ means that
Potential Infinity vs Actual Infinity (from Wikipedia)

- Actual Infinity is the idea that numbers, or some other type of
**mathematical object**, can form an actual, completed totality;- Such as the set of all natural numbers, an infinite sequence of rational numbers.

- Potential Infinity is a
**non-terminating process**(such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps.

- Actual Infinity is the idea that numbers, or some other type of
Precise Definitions

- $\displaystyle\lim_{x \to a}f(x) = L$ means:
- for all $\epsilon > 0$, there is $\delta>0$,
- so that if $0 < |x - a| < \delta$ ($x \ne a$ and x in within $\delta$ of
`a`

), then $|f(x)-L| < \epsilon$ (`f(x)`

is within $\epsilon$ of`L`

).`|x - a|`

is the distance from x to a and`|f(x) - L|`

is the distance from f(x) to L.- so $\displaystyle\lim_{x \to a}f(x) = L$ means that the distance between f(x) and L can be made arbitrarily small by requiring that the distance from x to a be sufficiently small (but not 0).

- For Example: $\displaystyle\lim_{x \to 2}x^2 = 4$
- Let's say $\epsilon = 0.1$, that means $|f(x)-4| < 0.1$,
`3.9 < f(x) < 4.1`

, - Base on the definition, there should be a $\delta$, that $2 - \delta < x < 2 + \delta$ to satisfy the demand.
- Try $\delta = 0.01$. We got
`1.99 < x < 2.01`

,`3.9601 < x^2 < 4.0401`

,`3.9 < f(x) < 4.1`

which suit the demand.- Another Example: $\displaystyle\lim_{x \to 10}2x = 20$
- Let $\epsilon > 0, \delta = \epsilon / 2$
- if $0<|x-10|<\delta$, then,
- $0<2|x-10|<2\delta=\epsilon$, and so,
- $0<|2x-20|<\epsilon$

- Let's say $\epsilon = 0.1$, that means $|f(x)-4| < 0.1$,

- $\displaystyle\lim_{x \to a}f(x) = L$ means: