Introductory maths analysis chapter 10 official

INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences

2007 Pearson Education Asia

Chapter 10 Chapter 10 Limits and ContinuityLimits and Continuity


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


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

INTRODUCTORY MATHEMATICAL ANALYSIS


• 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 ObjectivesChapter Objectives


Limits

Limits (Continued)

Continuity

Continuity Applied to Inequalities

10.1)

10.2)

10.3)


Chapter OutlineChapter Outline

10.4)



10.1 Limits10.1 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:

Lxfax

lim

2lim1

xfx

2lim1

xfx



10.1 Limits

Properties of Limits

1.

2. for any positive integer n

3.

4.

5.

constant a is wherelimlim cccxfaxax

nn

axax

lim

xgxfxgxfaxaxax

limlimlim

xgxfxgxfaxaxax

limlimlim

xfcxcfaxax

limlim



10.1 Limits

Example 3 – Applying Limit Properties 1 and 2

Properties of Limits

162lim c.

366lim b.

77lim ;77lim a.

44

2

22

6

52

t

x

t

x

xx

0lim if

lim

limlim 6.

xg

xg

xf

xg

xfax

ax

ax

ax

nax

n

axxfxf

limlim 7.



10.1 Limits

Example 5 – Limit of a Polynomial Function

Find an expression for the polynomial function,

Solution:

where

011

1 ... cxcxcxcxf nn

nn

af

cacacac

ccxcxc

cxcxcxcxf

nn

nn

axax

n

axn

n

axn

nn

nn

axax

011

1

011

1

011

1

...

lim lim...limlim

...limlim

afxfax

lim



10.1 Limits

Example 7 – Finding a Limit

Example 9 – Finding a Limit

Find .

Solution:

If ,find .Solution:

1

1lim

2

1

x

xx

2111lim1

1lim

1

2

1

x

x

xxx

12 xxf

h

xfhxfh

0

lim

xhxh

xhxhx

h

xfhxf

h

hh

22lim

112limlim

0

222

00

Limits and Algebraic Manipulation

• If f (x) = g(x) for all x a, then xgxf

axax limlim



10.2 Limits (Continued)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.

xx

1lim

0

xx

1lim

0

1

2lim a.

1 xx 4

2lim b.

22

x

xx



10.2 Limits (Continued)

Example 3 – Limits at Infinity


Solution:

a. b.

35

4lim a.

xx

0

5

4lim

3

xx

xx

4lim b.

xx

4lim

Limits at Infinity for Rational Functions

• If f (x) is a rational function,

and m

m

nn

xx xb

xaxf

limlim

mm

nn

xx xb

xaxf

limlim



10.2 Limits (Continued)

Example 5 – Limits at Infinity for Polynomial Functions


Solution:

Solution:

33 2lim92lim xxxxx

323 lim2lim xxxxxx

33 2lim92lim b. xxxxx

323 lim2lim a. xxxxxx



10.3 Continuity10.3 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:

afxf

xf

xf

ax

ax

lim 3.

exists lim 2.

exists 1.

55limlim77

xx

xf 75lim7

fxfx

43limlim 2

44

gxxg

xx



10.3 Continuity

Example 3 – Discontinuitiesa. 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

xf



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 xx

xxxf

2 if

2 if 2 b.

2 xx

xxxf



10.3 Continuity


Solution:

a. We know that f(3) = 3 + 6 = 9. Because

and , the function has no points of discontinuity.

96limlim33

xxfxx

9limlim 2

33

xxf

xx



10.3 Continuity


Solution:b. It is discontinuous at 2,

limx→2 f (x) exists.

xfxxxfxxxx

22

2

22lim2lim4limlim



10.4 Continuity Applied to Inequalities10.4 Continuity Applied to InequalitiesExample 1 – Solving a Quadratic Inequality

Solve .

Solution: Let .

To find the real zeros of f,

Therefore, x2 − 3x − 10 > 0 on (−∞,−2) (5,∞).

01032 xx

1032 xxxf

5 ,2

052

01032

x

xx

xx



10.4 Continuity Applied to Inequalities

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,∞).

0562

x

xx

x

xx

x

xxxf

51562

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