1. 3208 Unit 2 Limits and Continuity
Advanced Mathematics
3208
Unit 2
Limits and Continuity

2. NEED TO KNOW Expanding

3. Expanding
Expand the following:
A) (a + b)2
B) (a + b)3

4. C) (a + b)4

5. Pascals Triangle:

6. D) (x + 2)4
E) (2x -3)5

7. Look for Patterns
A) x2 – 9
B) x3 + 27

8. C) 8x3 - 64

9. 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 Graphs Using graphing technology:
Sketch the graph of y = x3 – 27

11. Analysis of Graphs 1. y = x3 – 27 A) Find the zeros
B) 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.

14. 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

15. 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

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.

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.

18. 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

19. Zeno’s Paradox Half of Halves Mathematically speaking:
This is the limit of an infinite series

20. How many sides does a circle have?
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
Note: This is not multiplication.

27. It does not matter if f(a) is defined.
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

28. Limits using a table of values.
1. Determine the behaviour of f (x) as x approaches 2.

29. Examples: (Using a Table of Values)
2. Find: x 3
2.5
2.1
2.01
2.001 x 1
1.5
1.9
1.99
1.999 5 3
4.5
3.5
This is the limit from the left side of x = 2
This is the limit from the right side of x = 2
4.1
3.9
4.01
3.99
4.001
3.999

30. Examples: (Using a Table of Values)
2.Find: q
(radians)
0.1
0.01
0.001 q
(radians)
-0.1
-0.01
-0.001

31. 3. For the function , complete the table below Sketch the graph of y = f(x) -5 -1 1 5
(x)

32. 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.

33. With reference to the previous graph complete the following table

34. One Sided Limits Consider the function below: This is a piecewise
It consists of two
different functions
combined together
into one function
What is the equation?

35. Find the following using the graph and function ruleB) C) D)
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
= 2
= 0
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.

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:

38. 2.A) Draw
B) Find:

39. 3.A) Draw
B) Find:
C) Find:

40. 4. Find

41. 5. Find

42. Evaluate the limits using the following piecewise function:

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

44. Text Page 33-34
3, 4, 7, 9, 15, 18

45. Absolute Values
Definition: The absolute value of a, |a|, is the distance a is from zero on a number line.
|3| = |-3| = |x| = 2
Note: - a is positive if a is negative

46. EX. |-5|
Here the value is negative so
|-5| = -(-5) = 5

47. Rewrite the following without absolute values symbols.1. 2. 3.

48. 4. |x + 2|

49. 5. |x| = 3
6. |x| < 3

50. 7. |x| > 3

51. Find
Recall:

53. Find

54. Greatest Integer Function
is the greatest integer function.
It gives the greatest integer that is less than or equal to x.
Example: A) B) D) C)

56. Find

59. HERE

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

61. 2. 3.

62. 4.
NOTE: : Direct substitution works in many cases, so you should always try it first.

63. 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.

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 ___
A) Draw the graph of
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.

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.

67. Examples 2. 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:

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??
Direct Substitution
Factor

69. 4.
What would you do here??
Direct Substitution
Factor

70. 5.
What would you do here??
Direct Substitution
More work!!

71. How do we rationalize a square root?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
Direct Substitution
Rationalize the
Numerator

73. 7.
What would you do here??
Simplify the
rational expression

74. 8. Find

75. 9. Find

76. 10. Find

77. Practice: A)

78. Practice: B)

79. Practice: C)

80. Practice: D)

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

82. Continuity

83. 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?

84. 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

85. Exponential and Logarithmic Functions
Sine and Cosine graphs
Absolute Value Graphs

86. What type of discontinuities are there?

87. 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

88. This can be broken into 3 parts
1. f(b) is defined (It exists)
b is in the domain of f(x)
exists.
In other words
3. Part 1 = Part 2

89. Describe why each place was discontinuous

90. Discuss the continuity of the following
1. 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?

91. 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
x = -1

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

93. 5. y = sinx 6. y = cos x 7. y = 2x Continuous everywhere
Not continuous at VA
7. y = 2x

94. Examples
What value of k would make the following functions continuous? 1.

97. 4. For what value of the constant c is the function
continuous at every number?

98. Page 54
# 1, 4, 7,15-18,31, 33,34

99. Page 27
# 1-5, 7, 9, 10

100. There is one other type of discontinuity
Graph
This is known
as an Oscillating
Discontinuity

101. 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.