Download Chapter 10 Limits and Continuity - Business Mathematics & Statistics

INTRODUCTORY MATHEMATICAL ANALYSIS
For Business, Economics, and the Life and Social Sciences
Chapter 10
Limits and Continuity
2007 Pearson Education Asia
INTRODUCTORY MATHEMATICAL ANALYSIS
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
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INTRODUCTORY MATHEMATICAL ANALYSIS
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
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Chapter 10: Limits and Continuity
Chapter Objectives
• 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.
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Chapter 10: Limits and Continuity
Chapter Outline
10.1) Limits
10.2) Limits (Continued)
10.3) Continuity
10.4) Continuity Applied to Inequalities
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Chapter 10: Limits and Continuity
10.1 Limits
• The limit of f(x) as x approaches a is the number L,
written as
lim f x   L
x a
Example 1 – Estimating a Limit from a Graph
a. Estimate limx→1 f (x) from the graph.
Solution: lim f x   2
x 1
b. Estimate limx→1 f (x) from the graph.
Solution: lim f x   2
x 1
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Chapter 10: Limits and Continuity
10.1 Limits
Properties of Limits
1. lim f x   lim c  c where c is a constant
x a
x a
2. lim x n  an for any positive integer n
x a

f x   g x   lim f x   lim g x 
3. lim
x a
x a
x a
4. limf x   g x   lim f x   lim g x 
x a
x a

cf x   c  lim f x 
5. lim
x a
x a
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x a
Chapter 10: Limits and Continuity
10.1 Limits
Properties of Limits
f x 
f x  lim
x a
6. lim

if lim g x   0
x a g x 
x a
lim g x 
x a
7. lim n f x   n lim f x 
x a
x a
Example 3 – Applying Limit Properties 1 and 2
a. lim 7  7;
lim 7  7
x 2
x 5
b. lim x 2  62  36
x 6
c. lim t 4   2  16
4
t 2
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Chapter 10: Limits and Continuity
10.1 Limits
Example 5 – Limit of a Polynomial Function
Find an expression for the polynomial function,
f x   cn x n  cn 1x n 1  ...  c1x  c0
Solution:

lim f x   lim c n x n  cn 1x n 1  ...  c1x  c0
x a
x a

 c n lim x n  cn 1 lim x n 1  ...  c1 lim  lim c0
x a
x a
 c n a n  cn 1a n 1  ...  c1a  c0
 f a 
where lim f x   f a 
x a
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x a
x a
Chapter 10: Limits and Continuity
10.1 Limits
Limits and Algebraic Manipulation
•
If f (x) = g(x) for all x  a, then
lim f x   lim g x 
x a
x a
Example 7 – Finding a Limit
Find
x2 1
.
lim
x 1 x  1
2
x
Solution: lim  1  lim x  1  1  1  2
x 1 x  1
x 1
Example 9 – Finding a Limit
f x  h   f x 
If f x   x  1 ,find h0
.
h
Solution: f x  h   f x 
x 2  2xh  h 2  1  x 2  1
lim
 lim
h 0
h 0
h
h
 lim2x  h   2x
2
lim

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h 0

Chapter 10: Limits and Continuity
10.2 Limits (Continued)
Infinite Limits
1
1
• Infinite limits are written as lim    and lim    .
x  0 x
x  0 x
Example 1 – Infinite Limits
Find the limit (if it exists).
2
a. lim
x  1 x  1
x2
b. lim 2
x 2 x  4
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.
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Chapter 10: Limits and Continuity
10.2 Limits (Continued)
Example 3 – Limits at Infinity
Find the limit (if it exists).
4
a. lim
x  x  5 3
b. lim
x 
4  x 
Solution:
4
0
x   x  5 3
a. lim
b. lim 4  x   
x 
Limits at Infinity for Rational Functions
• If f (x) is a rational function,
an x n
lim f x   lim
x 
x  b x m
m
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and
an x n
lim f x   lim
x 
x  b x m
m
Chapter 10: Limits and Continuity
10.2 Limits (Continued)
Example 5 – Limits at Infinity for Polynomial Functions
Find the limit (if it exists).


a. lim x 3  x 2  x  2  lim x 3
x 
x 
Solution: lim x 3  x 2  x  2  lim x 3  
x 

x 

b. lim  2x 3  9x  lim  2x 3
x 
x 
Solution: lim  2x 3  9x   lim  2x 3  
x 
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x 
Chapter 10: Limits and Continuity
10.3 Continuity
Definition
• f(x) is continuous if three conditions are met:
1. f x  exists
2. lim f x  exists
x a
3. lim f x   f a 
x a
Example 1 – Applying the Definition of Continuity
a. Show that f(x) = 5 is continuous at 7.
f x   lim 5  5, lim f x   5  f 7 .
Solution: Since lim
x 7
x 7
x 7
b. Show that g(x) = x2 − 3 is continuous at −4.
g x   lim x 2  3  g  4
Solution: xlim
4
x 4
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Chapter 10: Limits and Continuity
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.
 1 if x  0

b. Find discontinuity for f x    0 if x  0
 1 if x  0

Solution:
f is defined at x = 0 but limx→0 f (x) does not exist. f is
discontinuous at 0.
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Chapter 10: Limits and Continuity
10.3 Continuity
Example 5 – Locating Discontinuities in Case-Defined Functions
For each of the following functions, find all points of
discontinuity.
x  6 if x  3
a. f x   
2
x
if x  3

x  2 if x  2
b. f x   
2
x
if x  2

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Chapter 10: Limits and Continuity
10.3 Continuity
Example 5 – Locating Discontinuities in Case-Defined Functions
Solution:
a. We know that f(3) = 3 + 6 = 9. Because
lim f x   lim x  6   9 and lim f x   lim x 2  9 ,
x 3
x 3
x 3
x 3
the function has no points of discontinuity.


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
Chapter 10: Limits and Continuity
10.3 Continuity
Example 5 – Locating Discontinuities in Case-Defined Functions
Solution:
b. It is discontinuous at 2,
lim f x   lim x 2  4  lim x  2  lim f x 
x 2
x 2
limx→2 f (x) exists.
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x 2
x 2
Chapter 10: Limits and Continuity
10.4 Continuity Applied to Inequalities
Example 1 – Solving a Quadratic Inequality
Solve x 2  3 x  10  0 .
Solution: Let f x   x 2  3x  10.
To find the real zeros of f,
x 2  3 x  10  0
x  2x  5  0
x  2, 5
Therefore, x2 − 3x − 10 > 0 on (−∞,−2)  (5,∞).
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Chapter 10: Limits and Continuity
10.4 Continuity Applied to Inequalities
Example 3 – Solving a Rational Function Inequality
2
x
Solve  6 x  5  0 .
x
x 2  6 x  5 x  1x  5
Solution: Let f x  
.

x
x
The zeros are 1 and 5.
Consider the intervals: (−∞, 0) (0, 1) (1, 5) (5,∞)
Thus, f(x) ≥ 0 on (0, 1] and [5,∞).
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