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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
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
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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
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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
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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
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• The smaller the shaded area can be made, the better the approximation would be.
Guess what the function value is at x = 3
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Guess what the function value is at x = 3
25
Guess what the function value is at x = 3
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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.
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• 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
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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
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5. Find2
3
4, 3lim ( ) ( )
4, 3x
x xf x where f x
x x
Evaluate the limits using the following piecewise function:
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• 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
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58
• HERE
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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
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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
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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.
