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Gale BachFirst Course in Calculus
Math 1AFall 2015
http://online.santarosa.edu/homepage/gbach/
Why Math?
What is Math?
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
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