## 1.functions and limits

1-1 : Four ways to represent function

– function f $\rightarrow$ rule that assigns to each element x in a set D exactly one element, called f(x), in set E.

– Usually consider: D,E  sets of real numbers.

– D$\rightarrow$domain, and E$\rightarrow$range.

Representations of functions

1- verbally (words) 2- numerically (table) 3- visually (graph) 4- algebraically (formula).

Summetry

 function In algebra In geometry even f(-x)= f(x) Its graph is symmetric with respect to the y-axis Odd f(-x)= -f(x) Its graph is symmetric about the origin

Increasing and Decreasing functions

increasing:x1$<$x2 $\Leftrightarrow$f(x1$<$f(x2)

decreasing: x1$<$x2$< \Leftrightarrow$f(x1$>$f(x2)

1-2 : Mathematical Models : A Catalog of Essential Functions.

A mathematical model is mathematical description of real-world phenomenon, the model is to understand the phenomenon and perhaps to make predictions about future behavior.

There are many different types of functions that can be used to model relationships observed in the real word.

a- Linear Models $\mapsto y=f(x)=mx+b$
where: $m$  slope of the line, $b$  is the y– intercept.

b- Polynomials$\mapsto P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+...+a_{2}x^{2}+a_{1}x+a_{0}$

where:$a_{1},a_{2},a_{3}...a_{n}$  are constants called the coefficients of the polynomials. The domain of any polynomials is $\mathbb{R}=(-\infty,\infty )$, If  $a_{n}\neq 0$, then the degree of the polynomials is  $n$

c- Power Functions $\mapsto x^{a}$

where : $a$  is a constant

d- Rational Function $\mapsto f(x)=\frac{P(x)}{Q(x)}$

where :$P$ and $Q$  are polynomials, the domain consists of all values of x such that $Q(x)\neq 0$

e- Algebraic  Funcions: If it can be constructed using algebraic operations( such as additions, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. More example  $f(x)=\sqrt{x^{2}+1}$

f- Trigonometric Functions:$sin(x),cos(x),tan(x)...$

g- Exponential Functions $\mapsto f(x)=a^{x}$

where :$a$  is a positive constant.

h- Logarithmic Functions$\mapsto log_{a}x$

where:  the base $a$  is a positive constant.

1-3: New functions from old functions

Vertical and Horizontal Shifts: suppose  $c> 0$  . To obtain the graph of

• $y=f(x)+c$  shift the graph of  $y=f(x)$  a distance  $c$  units upward.
• $y=f(x)-c$  shift the graph of  $y=f(x)$  a distance  $c$  units downward
• $y=f(x-c)$  shift the graph of  $y=f(x)$  a distance  $c$  to the right
• $y=f(x+c)$  shift the graph of  $y=f(x)$  a distance  $c$  to the left

Vertical and Horizotal Stretching and Reflecting: Suppose  $c> 1$  To obtain the graph of

• $y=cf(x)$  stretch the graph of  $y=f(x)$  vertically by a factor of  $c$
• $y=(\frac{1}{c})f(x)$  shrink the graph of   $y=f(x)$  vertically by a factor of  $c$
• $y=f(cx)$  shrink the graph of   $y=f(x)$  horizontally by a factor of  $c$
• $y=f(\frac{x}{c})$  stretch the graph of  $y=f(x)$  horizontally by a factor of  $c$
•  $y=-f(x)$  reflect the graph of  $y=f(x)$  about x-axis
• $y=f(-x)$  reflect the graph of  $y=f(x)$  about y-axis

Combination of function

Definition: Given two functions  $f$  and  $g$  the composite function  $\displaystyle f\circ g$ (also called the composition of  $f$  and  $g$  ) is defined by  $\displaystyle (f\circ g)(x)=f(g(x))$

Note: in general  $\displaystyle f\circ g\neq g\circ f$

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1-5: The Limit of a Function

Definition (1): Suppose  $f(x)$  is defined when  $x$  is near the number  $a$  (This means that  $f$  is defined on some open interval that contains  $a$  , except possibly at  $a$  itself.) Then we write

$\lim_{x\to a}f(x)=\mathbf{L}$

and say “the limit of  $f(x)$  , as  $f(x)$  approaches  $a$  , equals   $\mathbf{L}$

if we can make the values of  $f(x)$   arbitrarily close to  $\mathbf{L}$  (as close to  $\mathbf{L}$  as we like) by taking  $x$  to be sufficiently close to  $a$  (on either side of  $a$  ) but not equal to  $a$

One- Sided Limits.

Definition (2): We write

$\mathbf{\lim_{x\to a^{-}}f(x)=L}$

and say the left- hand limit of  $\mathbf{f(x)}$  as  $x\mathbf{}$  approaches  $a\mathbf{}$  (or the limit of  $f(x\mathbf{})$  as  $x$  approaches  $a$  from the left ) is equal to  $L$  if we can make the values of  $f(x)$  arbitrarily close to   $L$  by taking   $x$  to be sufficiently close to   $a$  and  $x$  less than  $a$

Notice:that Definition (2) differs from Definition (1) only in that we require  $x$  to be less than  $a$  Similarly, if we require that  $x$  be greater than  $a$  we get “the right-hand limit of  $f(x)$  as  $x$  approaches  $a$  is equal to  $L$ ” and we write

$\lim_{x\to a^{+}}f(x)=L$

Thus the symbol “$x\rightarrow a^{+}$” means that wa consider only  $x> a$

3.  $\lim_{x\to a}f(x)=L$  if and only if  $\lim_{x\to a^{-}}f(x)=L$  and  $\lim_{x\to a^{+}}f(x)=L$

Infinite Limits

Definition (4): Let  $f$  be a function defined on both sides of  $a$  except possibly at  $a$  itself. Then

$\lim_{x\to a}f(x)=\infty$

means that the values of  $f(x)$  can be made arbitrarily large (as large as we please) by taking  $x$  sufficiently close to  $a$  but not equal to  $a$

Definition (5): Let  $f$  be defined on both sides of  $a$  except possibly at  $a$  itself. Then

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

means that the values of  $f(x)$  can be made arbitrarily large negative by taking  $x$  sufficiently close to  $a$  but not equal to  $a$

Definition (6): the line  $x=a$  is called vertical asymptote of the curve  $y=f(x)$  if at least one of following statements is true:

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

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

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1-6: Calculating Limits Using the Limit Laws

Limit Laws Suppose that $\displaystyle c$ is a constant and the limits

$\displaystyle \lim_{x\rightarrow a}f(x)$ and $\displaystyle \lim_{x\rightarrow a}g(x)$

exist. then

1$\displaystyle \lim_{x\rightarrow a}\left [f(x)+g(x) \right ]=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a}g(x)$

2$\displaystyle \lim_{x\rightarrow a}\left [f(x)-g(x) \right ]=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow a}g(x)$

3.$\displaystyle \lim_{x\rightarrow a}\left [cf(x)\right ]=c\lim_{x\rightarrow a}f(x)$

4.$\displaystyle \lim_{x\rightarrow a}\left [f(x)g(x) \right ]=\lim_{x\rightarrow a}f(x)\cdot\lim_{x\rightarrow a}g(x)$

5.$\displaystyle \lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow a}f(x)}{\lim_{x\rightarrow a} g(x)}$ if $\displaystyle \lim_{x\rightarrow a}g(x)\neq 0$

These five laws can be stated verbally as follows:

Sum Law 1. The limit of a sum is the sum of the limits

Difference Law 2. The limit of a difference is the difference of the limits

Constant Multiple Law 3. The limit of a constant times a function is the constant times the limit of the function

Product Law 4. The limit of a product is the product of the limits

Quotient Law 5. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0

Power Law 6$\displaystyle \lim_{x\rightarrow a}\left [ f(x) \right ]^{n}=\left [ \lim_{x\rightarrow a}f(x) \right ]^{n}$ where $\displaystyle n$ is a positive integer

7$\displaystyle \lim_{x\rightarrow a}c=c$

8. $\displaystyle \lim_{x\rightarrow a}x=a$

9$\displaystyle \lim_{x\rightarrow a}x^{n}=a^{n}$ where $\displaystyle n$ is a positive integer

10$\displaystyle \lim_{x\rightarrow a}\sqrt[n]{x}=\sqrt[n]{a}$ where $\displaystyle n$ is a positive integer (If $\displaystyle n$ is even, we assume that $\displaystyle a> 0$)

Root Law 11$\displaystyle \lim_{x\rightarrow a}\sqrt[n]{f(x)}=\sqrt[n]{\lim_{x\rightarrow a}f(x)}$ where $\displaystyle n$ is a positive integer

(If $\displaystyle n$ is even, we assume that $\displaystyle \lim_{x\rightarrow a}f(x)> a$)

Direct Substitution Property If $\displaystyle f$ is a polynomial or a rational function and $\displaystyle a$ is in the domain of $\displaystyle f$ then

$\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$

1.Theorem  $\displaystyle \lim_{x\rightarrow a}f(x)=L$ if and only if $\displaystyle \lim_{x\rightarrow a^{-}}f(x)=L=\lim_{x\rightarrow a^{+}}f(x)$

2. Theorem  If $\displaystyle f(x)\leqslant g(x)$ when $\displaystyle x$ is near $\displaystyle a$ (except possibly at $\displaystyle a$) and the limits of $\displaystyle f$ and $\displaystyle g$ both exist as $\displaystyle x$ approaches $\displaystyle a$ then

$\displaystyle \lim_{x\rightarrow a}f(x)\leqslant \lim_{x\rightarrow a}g(x)$

3. The Squeeze Theorem  If $\displaystyle f(x)\leqslant g(x)\leqslant h(x)$ when $\displaystyle x$ is near $\displaystyle a$ (except possibly at $\displaystyle a$) and
$\displaystyle \lim_{x\rightarrow a}f(x)=\lim_{x\rightarrow a} h(x)=L$
then $\displaystyle \lim_{x\rightarrow a}g(x)=L$

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1-7. The Precise Definition of a limit

2. Definition Let $f$ be a function on some open interval that contains the number $a$ except possibly at $a$ itself. Then we say that the limit of $f(x)$ as $x$ approaches $a$ is $L$, and we write

$\displaystyle \lim_{x\rightarrow a}f(x)=L$

if for every number $\epsilon > 0$ there is a number $\delta > 0$ such that
if $\displaystyle 0< \left |x-a \right |<\delta$ then $\displaystyle \left |f(x)-L \right |<\epsilon$

3. Definition of Left-Hand Limit

$\displaystyle \lim_{x\rightarrow a^{-}}f(x)=L$

if for every number $\displaystyle \epsilon > 0$ there is a number $\displaystyle \delta > 0$ such that

if $\displaystyle a-\delta then $\displaystyle \left | f(x)-L \right |< \epsilon$

4. Definition of Right-Hand Limit

$\displaystyle \lim_{x\rightarrow a^{+}}f(x)=L$

if for every number $\displaystyle \epsilon > 0$ there is a number $\displaystyle \delta> 0$ such that
if $\displaystyle a then $\displaystyle \left | f(x)-L \right |< \epsilon$

6- Definition Let $\displaystyle f$ be a function defined on some open interval that contains the number $\displaystyle a$ except possibly at $\displaystyle a$ itself. Then
$\displaystyle \lim_{x\rightarrow a}f(x)=\infty$
means that for every positive number $\displaystyle M$ there is a positive number $\displaystyle \delta$ such that
if $\displaystyle 0< \left | x-a \right |< \delta$ then $\displaystyle f(x)> M$

7. Definition let $\displaystyle f$ be a function defined on some open interval that contains the number $\displaystyle a$ except possibly at $\displaystyle a$ itself. then
$\displaystyle \lim_{x\rightarrow a}f(x)=-\infty$
means that for every negative number $\displaystyle N$ there is a positive number $\displaystyle \delta$ such that
if $\displaystyle 0< \left | x-a \right |< \delta$ then $\displaystyle f(x)< N$
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1-8: Continuity

1. Definition A function $\displaystyle f$ is continuous at a number $\displaystyle a$ if $\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$
Notice that Definition 1 implicitly requires three things if $\displaystyle f$ is continuous at $\displaystyle a$
1.$\displaystyle f(a)$ is defined (that is, $\displaystyle a$ is in the domain of $\displaystyle f$ )
2$\displaystyle \lim_{x\rightarrow a}f(x)$ exists
3$\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$

2. Definition A function $\displaystyle f$ is continuous from the right at a number $\displaystyle a$ if
$\displaystyle \lim_{x\rightarrow a^{+}}f(x)=f(a)$
and $\displaystyle f$ is continuous from the left at $\displaystyle a$ if
$\displaystyle \lim_{x\rightarrow a^{-}}f(x)=f(a)$

3. Definition A function $\displaystyle f$ is continuous on an interval if it is continuous at every number in the interval. (if $\displaystyle f$ is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left)

4. theorem If $\displaystyle f$ and $\displaystyle g$ are continuous at $\displaystyle a$ and $\displaystyle c$ is a constant, then the following functions are also continuous at $\displaystyle a$:
1$\displaystyle f+g$
2$\displaystyle f-g$
3$\displaystyle cf$
4$\displaystyle fg$
5$\displaystyle \frac{f}{g}$ if $\displaystyle g(x)\neq 0$

5. theorem

(a) Any polynomial is continuous everywhere; that is, it is continuous on  $\displaystyle \mathbb{R}=(-\infty ,\infty )$

(b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain

6$\displaystyle \lim_{\theta \rightarrow 0}cos \theta =1$   $\displaystyle \lim_{\theta \rightarrow 0}sin \theta =0$

7. theorem The following types of functions are continuous at every number in their domains:

polynomials    rational functions

root functions    trigonometric functions

8. theorem If $\displaystyle f$ is continuous at $\displaystyle b$ and $\displaystyle \lim_{x\rightarrow a}g(x)=b$ then $\displaystyle \lim_{x\rightarrow a}f(g(x))=f(b)$ In other words,
$\displaystyle \lim_{x\rightarrow a}f(g(x))=f(\lim_{x\rightarrow a}g(x))$

9. theorem If $\displaystyle g$ is continuous at $\displaystyle a$ and $\displaystyle f$ is continuous at $\displaystyle g(a)$, then the composite function $\displaystyle f\circ g$ given by $\displaystyle (f\circ g)(x)=f(g(x))$ is continuous at $\displaystyle a$

10. The Intermediate Value Theorem Suppose that $\displaystyle f$ is continuous on the closed interval $\displaystyle \left [ a,b \right ]$ and let $\displaystyle N$ be any number between $\displaystyle f(a)$ and $\displaystyle f(b)$, where $\displaystyle f(a)\neq f(b)$
Then there exists a number $\displaystyle c$ in $\displaystyle (a,b)$ such that $\displaystyle f(c)=N$

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stewart, 7E