Chapter 1 Differentiation

Chapter 1: Differentiation
Chapter 1
1.1
Differentiation
Limits: Numerical and Graphical Approach
OBJECTIVE: Find limits of functions, if they exist, using numerical or graphical methods.
DEFINITION
Limit
THEOREM
As B approaches +, the limit of 0 aBb is P, written
As B approaches +, the limit of 0 aBb
lim 0 aBb œ P
is P if the limit from the left exists and
BÄ+
if all values of 0 aBb are close to P for values of B
the limit from the right exists and both
limits are P. That is, if
that are sufficiently close, but not equal to, +.
1)
2)
lim 0 aBb œ P
BÄ+ •
and
lim 0 aBb œ P, then
BÄ+ €
lim 0 aBb œ P
BÄ+
1. If 0 aBb œ
B# • *
, what is 0 a$b?
B•$
2.
If 0 aBb œ
B# • *
, what is the limit
B•$
of 0 as B approachs $?
B Ä $•
#Þ!
#Þ&
#Þ*
B Ä $€
%Þ!
$Þ&
$Þ"
0 aBb
0 aBb
#Þ**
$Þ!"
#Þ***
$Þ!!"
1
2
Mr. Getso’s Business Calculus Class Notes
Consider the function L given by
#B € # for B • "
L aBb œ œ
#B • % for B "
Find each of the following limits numerically and graphically,
if they exist. When necessary, state that the limit does not exist.
3.
lim L aBb
4.
BÄ"
B Ä "•
!Þ!
B Ä "€
#Þ!
L aBb
L aBb
!Þ&
"Þ&
!Þ)
!Þ*
"Þ"
!Þ**
"Þ!"
lim L aBb
BÄ•$
!Þ***
B Ä $•
•%
•$Þ&
•$Þ"
•$Þ!"
•$Þ!!"
"Þ!!"
B Ä $€
•#
•#Þ&
•#Þ*
•#Þ**
•#Þ***
L aBb
L aBb
Chapter 1: Differentiation
3
The “Wall” Method:
As an alternative approach to the previous example, we can draw a “wall” at B œ ". Mr. Getso will draw
this “wall” on the following graphs. We then follow the curve from left to right with pencil until we hit
the wall and mark the location with an × , assuming it can be determined. Then we follow the curve
from right to left until we hit the wall and mark that location with an ×. If the locations are the same, we
have a limit. Otherwise, the limit does not exist.
lim L aBb does not exist.
lim L aBb œ •%
BÄ"
BÄ•$
Calculate the following limits based on the graph of 0 .
5. lim • 0 aBb
BÄ#
6.
lim € 0 aBb
BÄ#
7. lim 0 aBb
BÄ#
4
Mr. Getso’s Business Calculus Class Notes
Consider the function 0 given by
"
0 a Bb œ
€$
B•#
Find each of the following limits numerically and graphically,
if they exist. When necessary, state that the limit does not exist.
8.
lim 0 aBb
9.
BÄ$
lim 0 aBb
BÄ#
B Ä $•
#Þ"
#Þ&
#Þ*
#Þ**
B Ä #•
"Þ&
"Þ*
"Þ**
"Þ***
B Ä $€
$Þ&
$Þ#
$Þ"
$Þ!"
B Ä #€
#Þ&
#Þ"
#Þ!"
#Þ!!"
0 aBb
0 aBb
0 a Bb
0 a Bb
Chapter 1: Differentiation
10.
Consider the function 0 given by
"
0 a Bb œ
€$
B•#
Find lim 0 aBb.
BÄ_
Note: We can only approach _ from the left.
BÄ_
0 aBb
&
"!
"!!
"!!!
5
6
Mr. Getso’s Business Calculus Class Notes
Let 2aBb œ
"
€ '. Find the following limits:
"•B
11. lim 2aBb
12.
BÄ"
lim 2aBb
BÄ#
B Ä "•
!
!Þ&
!Þ*
!Þ**
!Þ***
B Ä #•
"Þ"
"Þ&
"Þ*
"Þ**
"Þ***
B Ä "€
#
"Þ&
"Þ"
"Þ!"
"Þ!!"
B Ä #€
$
#Þ&
#Þ"
#Þ!"
#Þ!!"
2aBb
2aBb
2 a Bb
2 a Bb
13. lim 2aBb
BÄ_
Note: We can only approach _ from the left.
BÄ_
2 a Bb
&
"!
"!!
"!!!

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