limitsandcontinuity-170512dsfsdsgssgs092805.ppt

INTRODUCTORY MATHEMATICAL ANALYSIS
For Business, Economics, and the Life and Social Sciences
2007 Pearson Education Asia
Chapter 10
Chapter 10
Limits and Continuity
Limits and Continuity

0. Review of Algebra
1. Applications and More Algebra
2. Functions and Graphs
3. Lines, Parabolas, and Systems
4. Exponential and Logarithmic Functions
5. Mathematics of Finance
6. Matrix Algebra
7. Linear Programming
8. Introduction to Probability and Statistics

9. Additional Topics in Probability
10. Limits and Continuity
11. Differentiation
12. Additional Differentiation Topics
13. Curve Sketching
14. Integration
15. Methods and Applications of Integration
16. Continuous Random Variables
17. Multivariable Calculus

• To study limits and their basic properties.
• To study one-sided limits, infinite limits, and
limits at infinity.
• To study continuity and to find points of
discontinuity for a function.
• To develop techniques for solving nonlinear
inequalities.
Chapter 10: Limits and Continuity
Chapter Objectives
Chapter Objectives

Limits
Limits (Continued)
Continuity
Continuity Applied to Inequalities
10.1)
10.2)
10.3)
Chapter Outline
Chapter Outline
10.4)

10.1 Limits
10.1 Limits
Example 1 – Estimating a Limit from a Graph
• The limit of f(x) as x approaches a is the number L,
written as
a. Estimate limx→1 f (x) from the graph.
Solution:
b. Estimate limx→1 f (x) from the graph.
Solution:
  L
x
f
a
x


lim
  2
lim
1


x
f
x
  2
lim
1


x
f
x

10.1 Limits
Properties of Limits
1.
2. for any positive integer n
3.
4.
5.
  constant
a
is
where
lim
lim c
c
c
x
f
a
x
a
x




n
n
a
x
a
x 

lim
   
     
x
g
x
f
x
g
x
f
a
x
a
x
a
x 




 lim
lim
lim
   
     
x
g
x
f
x
g
x
f
a
x
a
x
a
x 




 lim
lim
lim
 
   
x
f
c
x
cf
a
x
a
x 


 lim
lim

10.1 Limits
Example 3 – Applying Limit Properties 1 and 2
Properties of Limits
  16
2
lim
c.
36
6
lim
b.
7
7
lim
;
7
7
lim
a.
4
4
2
2
2
6
5
2












t
x
t
x
x
x
 
 
 
 
  0
lim
if
lim
lim
lim
6. 





x
g
x
g
x
f
x
g
x
f
a
x
a
x
a
x
a
x
   
n
a
x
n
a
x
x
f
x
f


 lim
lim
7.

10.1 Limits
Example 5 – Limit of a Polynomial Function
Find an expression for the polynomial function,
Solution:
where
  0
1
1
1 ... c
x
c
x
c
x
c
x
f n
n
n
n 



 

   
 
a
f
c
a
c
a
c
a
c
c
c
x
c
x
c
c
x
c
x
c
x
c
x
f
n
n
n
n
a
x
a
x
n
a
x
n
n
a
x
n
n
n
n
n
a
x
a
x




























0
1
1
1
0
1
1
1
0
1
1
1
...
lim
lim
...
lim
lim
...
lim
lim
   
a
f
x
f
a
x


lim

10.1 Limits
Example 7 – Finding a Limit
Example 9 – Finding a Limit
Find .
Solution:
If ,find .
Solution:
1
1
lim
2
1 

 x
x
x
  2
1
1
1
lim
1
1
lim
1
2
1













x
x
x
x
x
  1
2

x
x
f
   
h
x
f
h
x
f
h


 0
lim
     
  x
h
x
h
x
h
xh
x
h
x
f
h
x
f
h
h
h
2
2
lim
1
1
2
lim
lim
0
2
2
2
0
0














Limits and Algebraic Manipulation
• If f (x) = g(x) for all x  a, then
   
x
g
x
f
a
x
a
x 

lim
lim

10.2 Limits (Continued)
Example 1 – Infinite Limits
Infinite Limits
• Infinite limits are written as and .
Find the limit (if it exists).
Solution:
a. The results are becoming arbitrarily large. The limit
does not exist.
b. The results are becoming arbitrarily large. The limit
does not exist.




 x
x
1
lim
0




 x
x
1
lim
0
1
2
lim
a.
1 


 x
x 4
2
lim
b. 2
2 

 x
x
x

Example 3 – Limits at Infinity
Solution:
a. b.
 3
5
4
lim
a.


 x
x
 
0
5
4
lim 3



 x
x
 
x
x



4
lim
b.
  




x
x
4
lim
Limits at Infinity for Rational Functions
• If f (x) is a rational function,
and
  m
m
n
n
x
x x
b
x
a
x
f




lim
lim   m
m
n
n
x
x x
b
x
a
x
f




 lim
lim

Example 5 – Limits at Infinity for Polynomial Functions
Solution:
Solution:   









3
3
2
lim
9
2
lim x
x
x
x
x
  









3
2
3
lim
2
lim x
x
x
x
x
x
  3
3
2
lim
9
2
lim
b. x
x
x
x
x








  3
2
3
lim
2
lim
a. x
x
x
x
x
x 








10.3 Continuity
10.3 Continuity
Example 1 – Applying the Definition of Continuity
Definition
• f(x) is continuous if three conditions are met:
a. Show that f(x) = 5 is continuous at 7.
Solution: Since , .
b. Show that g(x) = x2
− 3 is continuous at −4.
Solution:
 
 
   
a
f
x
f
x
f
x
f



a
x
a
x
lim
3.
exists
lim
2.
exists
1.
  5
5
lim
lim
7
7



 x
x
x
f    
7
5
lim
7
f
x
f
x



     
4
3
lim
lim 2
4
4








g
x
x
g
x
x

10.3 Continuity
Example 3 – Discontinuities
a. When does a function have infinite
discontinuity?
Solution:
A function has infinite discontinuity at a when at least
one of the one-sided limits is either ∞ or −∞ as x →a.
b. Find discontinuity for
Solution:
f is defined at x = 0 but limx→0 f (x) does not exist. f is
discontinuous at 0.
 










0
if
1
0
if
0
0
if
1
x
x
x
x
f

10.3 Continuity
Example 5 – Locating Discontinuities in Case-Defined Functions
For each of the following functions, find all points of
discontinuity.
 







3
if
3
if
6
a. 2
x
x
x
x
x
f
 







2
if
2
if
2
b. 2
x
x
x
x
x
f

10.3 Continuity
Solution:
a. We know that f(3) = 3 + 6 = 9. Because
and ,
the function has no points of discontinuity.
    9
6
lim
lim
3
3


 



x
x
f
x
x
  9
lim
lim 2
3
3




 
x
x
f
x
x

10.3 Continuity
Solution:
b. It is discontinuous at 2,
limx→2 f (x) exists.
   
x
f
x
x
x
f
x
x
x
x 












2
2
2
2
2
lim
2
lim
4
lim
lim

10.4 Continuity Applied to Inequalities
Example 1 – Solving a Quadratic Inequality
Solve .
Solution: Let .
To find the real zeros of f,
Therefore, x2
− 3x − 10 > 0 on (−∞,−2)  (5,∞).
0
10
3
2


 x
x
  10
3
2


 x
x
x
f
  
5
,
2
0
5
2
0
10
3
2








x
x
x
x
x

Example 3 – Solving a Rational Function Inequality
Solve .
Solution: Let .
The zeros are 1 and 5.
Consider the intervals: (−∞, 0) (0, 1) (1, 5) (5,∞)
Thus, f(x) ≥ 0 on (0, 1] and [5,∞).
0
5
6
2



x
x
x
    
x
x
x
x
x
x
x
f
5
1
5
6
2







limitsandcontinuity-170512dsfsdsgssgs092805.ppt

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