Gale BachFirst Course in Calculus

Math 1AFall 2015

http://online.santarosa.edu/homepage/gbach/

Why Math?

What is Math?

Jobs?

Thales was the first known Greek philosopher, scientist and mathematician. He is credited with five theorems of elementary geometry.

Thales of Miletus625 – 547 B.C.

The Ancients

2Consider the function ( ) 5 6.f x x x

As x approaches 2, what is the behavior of f (x)?

There will be three strategies for analyzing this question.

1.) Graphically

2.) Numerically

3.) Algebraically/Symbolically

In section 2.2, our analysis will be graphical and numerical,the warm and fuzzy way!

Example 1

2 ( ) 5 6, so what happens to ( ) as approaches 2?f x x x f x x 2

2We write this question mathematically lim 5 6.

xx x

2

2

So we say

lim 5 6 12.x

x x

The limit of a function refers to the value that the function approaches, not the actual value (if any).

2

limx

f x

2

321

1

2 y f x

Solution

0

sinSo, lim 1.

x

x

x

Example 2

Hole in the graph

Properties of Limits:

For a limit to exist, the function must approach the same value from both sides.

One-sided limits or directional limits approach from either the left or right side only.

Example 3

Properties of Limits:

For a limit to exist, the function must approach the same value from both sides.

One-sided limits or directional limits approach from either the left or right side only.

1

lim ?x

f x

Example 3

So, lets consider the function f (x) below.

1Find lim ( ).

xf x

1

lim ( )x

f x

4Limit from the left

Limit from the right

1lim ( )x

f x

6

These are called one-sidedor directional limits.

1So what is lim ( )?

xf x

From the left

From the right

1 1

Since,

lim ( ) lim ( )x x

f x f x

1

We say,

lim ( ) Does Not Existx

f x

1

Important Note,

lim ( ) Does Not Exist.

But,

(1) 3.

xf x

f

1 2 3 4

1

2

At x = 1: 1

limx

f x

1

limx

f x

1f

left hand limit

right hand limit

value of the function

1

limx

f x

does not exist

because the left and right hand limits do not match!

0

1

1

y

x

( )y f x

Example 4

At x = 2: 2

limx

f x

2

limx

f x

2f

left hand limit

right hand limit

value of the function

2

lim 1x

f x

because the left and right hand limits match.

1 2 3 4

1

2

1

1

2

( )y f xy

x

Example 4

At x = 3: 3

limx

f x

3

limx

f x

3f

left hand limit

right hand limit

value of the function

3

lim 2x

f x

because the left and right hand limits match.

1 2 3 4

1

2

2

2

2

All three are equal! I wonder if that is something special?

( )y f xy

x

Example 4

23

3Find the lim .

9x

x

x

Example 5

23

23

It appears that

3 1 lim

9 6and

3 1 lim .

9 6

x

x

x

x

x

x

What is (3)?f

Does Not Exist.

23

Thus the

nondirectional,

3 1 lim .

9 6x

x

x

23

3Find the lim .

9x

x

x

Example 5

23

23

It appears that

3 1 lim

9 6and

3 1 lim .

9 6

x

x

x

x

x

x

What is (3)?f

Does Not Exist.

23

Thus the

nondirectional,

3 1 lim .

9 6x

x

x

1Hole at 3,

6

23

3Find the lim .

9x

x

x

Example 5

0

1lim cos does not exist.x x

1( ) cosf x

x

0

1 1Consider the function ( ) cos . Find the lim cos .

xf x

x x

Example 6

= 4DNE

Example 7 Find each limit.

Quick Quiz

Example 8

Example 9

Example 10

Example 11

Example 12