advanced mathematics 3208 unit 2 limits and continuity

Advanced Mathematics

Advanced Mathematics

3208Unit 2

Limits and Continuity

NEED TO KNOW Expanding

NEED TO KNOW Expanding

Expanding• Expand the following:A) (a + b)2

B) (a + b)3

C) (a + b)4

Pascals Triangle:

D) (x + 2)4

E) (2x -3)5

Look for PatternsA) x2 – 9

B) x3 + 27

C) 8x3 - 64

9

II. Functions, Graphs, and Limits

II. Functions, Graphs, and Limits

Analysis of graphs. •With the aid of technology.•Prelude to the use of calculus both to predict and to explain the observed local and global behaviour of a function.

10

Analysis of GraphsUsing graphing technology:1. Sketch the graph of y = x3 – 27

11

Analysis of Graphs1. y = x3 – 27A) Find the zerosB) Find the local max and min

points• These are points that have either the

largest, or smallest y value in a particular region, or neighbourhood on the graph.

x = 3

• There are no local max or min points

12

C) Identify any points where concavity changes from concave up to concave down (or vice a versa).

The point of inflection is (0, -27)

13

2. Sketch the graph of:A)

B) y = x – 2

What do you notice?• y = x – 2 is a slant (or oblique)

asymptote.

2 12

xy

x

Rational Functions• f(x) is a rational function if

where p(x) and q(x) are polynomials and

• Rational functions often approach either slant or horizontal asymptotes for large (or small) values of x

( )( )

( )p x

f xq x

( ) 0q x

• Rational Functions are not continuous graphs.

• There various types of discontinuities.– There vertical asymptotes which occur

when only the denominator (bottom) is zero.

– There are holes in the graph when there is zero/zero

00

16

3. Describe what happens to the function near x = 2.– The graph seems to approach the point

(2, 4)• What occurs at x = 2?

– Division by zero. The function is undefined when x = 2. In fact we get

– There is a hole in the graph.• What occurs at x = -2?

–Division by zero however this time there is a vertical asymptote.

2

24

xy

x

00

17

4. Describe what happens to the function as x gets close to 0.

• The function seems to approach 1

• Does it make any difference if the calculator is in degrees or radians?

• Yes, it only approaches 1 in radians.

sinxy

x

Limits of functions (including one-sided limits).

Limits of functions (including one-sided limits).

•A basic understanding of the limiting process.•Estimating limits from graphs or tables of data.•Calculating limits using algebra.•Calculating limits at infinity and infinite limits

Zeno’s Paradox• Half of Halves

• Mathematically speaking:

• This is the limit of an infinite series

19

1 1 1 1 1...

2 4 8 16 32

12i

1

12

n

ii

1

1lim

2

n

in i

• How many sides does a circle have?

http://www.mathopenref.com/circleareaderive.html

20

5 sides? 18 sides?

21

Limit of a Function• The limit of a function tells how a

function behaves near a certain x-value.

• Suppose if I wanted to go to a certain place in Canada.

• We would use a map

22

Consider:• If we have a function

y = f(x) and we are trying to find out what the value of the function is for a x-value under the shaded area, we could make an estimate of what it would be by looking at the function before it goes into or leaves the shaded area.

Guess what the function value is at x = 3

23

• The smaller the shaded area can be made, the better the approximation would be.

Guess what the function value is at x = 3

24

Guess what the function value is at x = 3

25

Guess what the function value is at x = 3

26

Mathematically speaking:

• As x gets close to a, f(x) gets close to a value L

• This can be written:

• It means “The limit of f(x) as x approaches a equals L

lim ( )x a

f x L

Note: This is not multiplication.

27

• We can get values of f(x) to be arbitrarily close to L by looking at values of x sufficiently close to a, but not equal to a.

• It does not matter if f(a) is defined.• We are only looking to see what happens

to f(x) as x approaches a

Limits using a table of values.

1. Determine the behaviour of f (x) asx approaches 2.

28

29

Examples: (Using a Table of Values)

2. Find:2

2

4lim

2x

xx

x

3

2.5

2.1

2.01

2.001

2 42

xx

5

4.5

4.01

4.1

4.001

x

1

1.5

1.9

1.99

1.999

2 42

xx

3

3.5

3.9

3.99

3.999

2

2

4lim 4

2x

xx

2

2

4lim 4

2x

xx

2

2

4lim 4

2x

xx

Th

is is

the lim

it f

rom

th

e r

ight

side o

f x =

2

Th

is is

the lim

it f

rom

th

e left

sid

e o

f x =

2

30

Examples: (Using a Table of Values)

2.Find:0

sinlim

q(radians)

0.1

0.01

0.001

sin

0.998334

q(radians)

-0.1

-0.01

-0.001

sin

0

sinlim 1

0

sinlim 1

0

sinlim 1

0.9999998

0.998334

0.9999833 0.9999833

0.9999998

3. For the function , complete the table below

Sketch the graph of y = f(x)

31

x -5 -1 0 1 5

(x)

12

14

14

12

1( )f x

x

x

y

x

y

Using the table and graph as a guide, answer following questions:

• What value is f (x) approaching as x becomes a larger positive number?

• What value is f (x) approaching as x becomes a larger negative number?

• Will the value of f (x) ever equal zero? Explain your reasoning.

32

With reference to the previous graph complete the following table

33

34

One Sided LimitsConsider the function below:

This is a piecewisefunction

It consists of twodifferent functions combined together into one function

What is the equation?

2 1, 1( )

1, 1

x xf x

x x

35

Find the following using the graph and function

ruleA)

B)

C)

D)

1lim ( )x

f x

0lim ( )x

f x

2lim ( )x

f x

1lim ( )x

f x

For this limit we need to find both the left and right hand limits because the function has different rules on either side of 1.

36

• In this case we say that the limit Does Not Exist – (DNE)

• NOTE: Limits do not exist if the left and right limits at a x-value are different.

1lim ( )x

f x

2

1lim 1x

x

1lim ( )x

f x

1lim 1x

x

= 0 = 2

37

Mathematically Speaking

• A function will have a limit L as x approaches a, if and only if as x approaches a from the left and a from the right you get the same value, L.

• OR:lim ( )x a

f x L

lim ( )x a

f x L

lim ( )x a

and

f x L

( )iff

38

2.A) Draw

B) Find:

2

2, 2( )

,1 2

x xf x

x x

2lim ( )x

f x

39

3.A) Draw 2 2, 1

( )2, 1

x xf x

x x

0lim ( )x

f x

B) Find:

C) Find:

1lim ( )x

f x

40

4. Find 1

2, 1lim ( ) ( )

2, 1x

x xf x where f x

x

41

5. Find2

3

4, 3lim ( ) ( )

4, 3x

x xf x where f x

x x

Evaluate the limits using the following piecewise function:

42

• Identify which limit statements are true and which are false for the graph shown.

43

• Text Page 33-34• 3, 4, 7, 9, 15, 18

44

Absolute Values• Definition: The absolute value of a,

|a|, is the distance a is from zero on a number line.|3| = |-3| = |x| = 2

| |

|

0

0|

a if

a a

a a

aif

Note: - a is positive if a is negative

• EX. |-5| – Here the value is negative so

• |-5| = -(-5) = 5

Rewrite the following without absolute values symbols.

1.

2.

3.

| 3|

| 3 |

| |x

4. |x + 2|

2 2 0

( 2) 2 0

x if x

x if x

2 2

( 2) 2

x if x

x if x

5. |x| = 3

6. |x| < 3

3 0

3 0

x if x

x if x

7. |x| > 3

51

Find

Recall:

0

| |limx

xx

, 0|

,|

0

x if x

x if xx

0 0

| |lim lim 1x x

x xx x

0 0

| |lim lim 1x x

x xx x

0

| |limx

xDNE

x

52

53

Find

Find

1

| 1|lim

1x

xx

22

| 2|lim

4x

xx

54

Greatest Integer Function

is the greatest integer function.• It gives the greatest integer that is

less than or equal to x.• Example:A)

x

x

2 B)

C) D)

2.2

2.99 0.2

55

56

Find 2

limx

x

2

limx

x

2

limx

x

2

limx

x DNE

57

58

• HERE

59

60

Solving Limits Using Algebra

• There are 7 limit laws which basically allow you to do direct substitution when finding limits.

• Examples:Evaluate and justify each step by indicating the appropriate Limit Law

3

1. lim 2 1x

x

61

0lim ( 1)x

x x

2.

3.

2

1lim2 3 1x

x x

4.

NOTE: : Direct substitution works in many cases, so you should always try it first.

62

3

2

2lim

5 2x

x xx

NOTE: These limit laws basically allow you to do Direct Substitution.

4.

• Direct Substitution works in many cases, so you should always try it first.

63

3

2

2lim

5 2x

x xx

64

• However, there are a few cases (mostly in math courses) where direct substitution does not work immediately, or at all.

65

• In this case direct substitution would give an answer of ___– which is not correct.

• Remember the limit shows what the function is approaching as x approaches a value.

• It does not matter what the actual function value is at that x value.

2, 3( )

2, 3

x xf x

x

A) Draw the graph of

x

y

3B) Find lim ( )

xf x

66

Examples

1.Direct substitution gives

which is undefined.• In this case the limit will not work

because the x value the limit is approaching is not in the domain of the function.

1limx

x

1

1

limx

x DNE Does Not Exist

67

Examples2.

Direct substitution gives which is undefined.

• In this case direct substitution will not work because the x value the limit is approaching is not in the domain of the function.

• However, as we will see later this one would not be DNE. Here we say that:

20

1limx x

10

20

1limx x

68

3.

• Whenever you get , this means there is some simplification you can do to the function before you do the direct substitution.

What would you do here??

2

2

4lim

2x

xx

22 4 02 2 0

Direct Substitution

00

Factor2

2 2

4 ( 2)( 2)lim lim

2 2x x

x x xx x

2lim 2 2 2 4x

x

69

4.

What would you do here??

2

21

2lim

2 1x

x xx x

Direct Substitution

Factor

2

2

1 1 22(1) 1 1

00

70

5.

What would you do here??

20

2 4limh

h

h

Direct Substitution

More work!!

22 0 4

0

00

71

6.

What would you do here??How do we rationalize a square root?

• We multiply top and bottom by the conjugate.

• The conjugate is the other factor of the difference of squares

2

2 2lim

2x

xx

Direct Substitution

Rationalize theNumerator

2 2 42 2

00

72

2

2 2lim

2x

xx

73

7.

What would you do here??

1

0

12

2limh

h

h

Simplify the rational expression

1 1 1 12 0

2 2 20 0

00

74

8. Find

4

2 2 4lim

4x

x x x

x

75

9. Find 1

3 3lim

1x

x x x xx

76

10. Find 2

3 6lim

2 2x

xx x

77

Practice:A)

2

23

2 3lim

6x

x xx x

78

Practice:B)

3

21

1lim

1x

xx

79

Practice:C)

9

9lim

3x

xx

80

Practice:D)

3

13 4lim

7 2x

xx

81

Page 44-45# 3, 11, 14, 15, 17-19,21-28, 44,45

Continuity

What is meant by a continuous function?

• A curve that can be drawn without taking your pencil from the paper.

• Which letters of the alphabet are the result of continuous lines?

What functions are continuous?• Polynomials

• These are continuous everywhere

• Rational Functions • These are continuous for all values of x

except for the roots of g(x) = 0.• In other words it is continuous for all

values in the domain

( )( )

f xg x

• Exponential and Logarithmic Functions

• Sine and Cosine graphs

• Absolute Value Graphs

What type of discontinuities are there?

• We need a way of defining continuity to know whether or not a function is discontinuous or continuous at a point.

• Definition: A function y = f(x) is continuous at a number b, if

lim ( ) ( )x b

f x f b

This can be broken into 3 parts

1. f(b) is defined (It exists) • b is in the domain of f(x)

2. exists.• In other words

3. Part 1 = Part 2

lim ( )x b

f x lim ( ) lim ( )

x b x bf x f x

lim ( ) ( )x b

f x f b

Describe why each place was discontinuous

Discuss the continuity of the following1. f(x) = x3 + 2x + 1

• This is continuous everywhere because it is a polynomial.

• Discontinuous at x = 1 (VA) • 1 is not in the Domain

• Not continuous at x = 3. WHY?

2. ( )1

xg x

x

4, 33. ( )

2, 3

x xh x

x

3lim ( ) 3 4 7 (3) 2x

f x f

• We need to check x = -1 and x = 1.• Do we need to check x = 0?

– NO! In 1/x, x=0 is not in x < -1

• Thus f(x) is continuous at x = 1

2

1, 1

4. ( ) , 1 1

1, 1

xx

f x x x

x x

1lim ( )

xf x

1

1lim

x x

1

x = -1

1lim ( )

xf x

2

1lim

xx

1

• Thus f(x) is discontinuous at x = 1 since the left and right limits are not the same.

2

1, 1

4. ( ) , 1 1

1, 1

xx

f x x x

x x

1lim ( )x

f x

2

1limx

x

2

x = 1

1lim ( )x

f x 1

lim 1x

x

1

5. y = sinx• Continuous everywhere

6. y = cos x• Not continuous at VA

7. y = 2x • Continuous everywhere

,2

x k k

Examples

What value of k would make the following functions continuous?

1.

2 4, 2

( ) 2, 2

xx

f x xk x

2 2 , 22. ( )

5 , 2

x x xh x

x k x

2 , 13. ( )

3, 1

x kx xf x

kx x

4. For what value of the constant c is the function

continuous at every number?

2

, 2( )

1, 2

x c xf x

cx x

• Page 54

# 1, 4, 7,15-18,31, 33,34

• Page 27

# 1-5, 7, 9, 10

There is one other type of discontinuity • Graph

• This is known

as an Oscillating

Discontinuity

1siny

x

x

y

• The function sin(1/x) is not defined

at x = 0 so it is not continuous at

x = 0. • The function also oscillates

between -1 and 1 as x approaches 0. – Therefore, the limit does not exist.