Transcript II. Functions, Graphs, and Limits (3207)

Advanced Mathematics
3208
Unit 2
Limits and Continuity
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 Patterns
A) x2 – 9
B) x3 + 27
C) 8x3 - 64
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.
9
Analysis of Graphs
Using graphing technology:
1. Sketch the graph of y = x3 – 27
10
Analysis of Graphs
1. y = x3 – 27
A) Find the zeros x = 3
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.
•There are no local max or min points
11
C) Identify any points where
concavity changes from concave up to
concave down (or vice a versa).
The point of inflection is (0, -27)
12
2. Sketch the graph of:
x2 1
A) y 
x 2
B)
y=x–2
What do you notice?
• y = x – 2 is a slant (or oblique)
asymptote.
13
Rational Functions
p (x )
• f(x) is a rational function if f (x ) 
q (x )
where p(x) and q(x) are polynomials
and q (x )  0
• Rational functions often approach
either slant or horizontal asymptotes
for large (or small) values of 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 0
 
 
0
3. Describe what happens to the
x 2
function y  x 2  4 near x = 2.
– The graph seems to approach the point
(2, 4)
• What occurs at x = 2?
– Division by zero. The function is
0
undefined when x = 2. In fact we get  
0
– There is a hole in the graph.
• What occurs at x = -2?
– Division by zero however this time there is
a vertical asymptote.
16
4. Describe what happens to the
sin x
function y 
as x gets close to
x
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.
17
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
1 1 1
1
1
  

 ... 
2 4 8 16 32
• Mathematically speaking:
11
i
lim

2
n i 1 2i
nn
i 1
• This is the limit of an infinite series
19
• How many sides does a circle have?
5 sides?
18 sides?
http://www.mathopenref.com/circleareaderive.html
20
Limit of a Function
• The limit of a function tells how a
function behaves near a certain xvalue.
• Suppose if I wanted to go to a
certain place in Canada.
• We would use a map
21
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 xvalue 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
22
• The smaller the
shaded area can be
made, the better
the approximation
would be.
Guess what the function value is at x = 3
23
Guess what the function value is at x = 3
24
Guess what the function value is at x = 3
25
Mathematically speaking:
• As x gets close to a, f(x) gets close
to a value L
f (x )  L
• This can be written: xlim
a
Note: This is not multiplication.
• It means “The limit of f(x) as x
approaches a equals L
26
• 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
27
Limits using a table of values.
1. Determine the behaviour of f (x) as
x approaches 2.
28
3
5
2.5
4.5
2.1
4.1
2.01
4.01
2.001
4.001
x2 4
lim
4
x 2 x  2

This is the limit from the left side of x = 2
2
This is the limit from the right side of x = 2
Examples: (Using a Table of Values)
2
x2 4
x
4
lim
2. Find:
 lim
4
x 2 x  2
x 2 x  2
x 4
x 4
x
x
x 2
2
x 2
1
3
1.5
3.5
1.9
3.9
1.99
3.99
1.999
3.999
x2 4
lim
4
x 2 x  2

29
Examples: (Using a Table of Values)
sin 
sin 
lim
 lim
1
2.Find:  0
 0 

sin 
sin 



(radians)
0.1
0.998334

(radians)
-0.1
0.998334
0.01
0.9999833
-0.01
0.9999833
0.001
0.9999998
-0.001
0.9999998
lim
 0
sin 

1
lim
 0
sin 

1
30
3. For the function
table below
x
-5
-1
(x)

1
2
f (x ) 

1
x
0
1
4
, complete the
1
4
1
1
2
5
Sketch the graph of y = f(x)
y


x






31
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
One Sided Limits
Consider the function below:

This is a piecewise
function
It consists of two
different functions
combined together
into one function
















What is the equation?
2


x
 1, x  1

f (x )  

x  1, x  1
34
Find the following using the
graph and function rule
f (x )
A) xlim
1
f (x )
B) xlim
0
f (x )
C) lim
x 2
D)
lim f (x )
x 1
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.
35
lim f (x )
lim f (x )
x 1
x 1
x 1
x 1
lim  x 2  1
=0
lim x  1
=2
• 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.
36
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 f (x )  L
lim f (x )  L
(
iff
)

x a
x a

and
lim f (x )  L
x a
37
x  2, x  2
f (x )   2
x  1, x  2
2.A) Draw




























f (x )
B) Find: lim
x 2
38
3.A) Draw
x 2  2, x  1
f (x )  
x  2, x  1


B) Find:
lim f (x )

x 0


C) Find:

lim f (x )














x 1








39
4. Find
x  2, x  1
limf (x ) where f (x )  
x 1
2, x  1

















40
5. Find
2

x
  4, x  3
lim f (x ) where f (x )  
x 3

x  4, x  3

















41
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










| a | a if a  0


| a | a if a  0
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. |   3|
2. |3   |
3. | x |
4.
|x + 2|
x  2 if x  2  0

(x  2) if x  2  0
x  2 if x  2

(x  2) if x  2
5.
|x| = 3
x  3 if x  0

x  3 if x  0
6.
|x| < 3
7.
|x| > 3
Find
lim
|x |
x
x 0
x , if x  0
Recall: | x | 
x , if x  0
lim
x 0
|x |
x
 lim
x 0
x
x
x
lim
 lim
1
x 0
x 0 x
x
 1
 lim
x 0
|x |

|x |
x

 DNE
51

















52
Find
Find
| x  1|
lim
x 1 x  1
| x  2|
lim 2
x 2 x  4
53
x
Greatest Integer Function
x
is the greatest integer function.
• It gives the greatest integer that is
less than or equal to x.
• Example:
B) 2.2
A) 2
C)
2.99
D)
0.2
54

















55
Find
lim x
x 2
lim x 
lim x 
x 2
x 2
 lim x  DNE
x 2
56
57
58
• HERE
59
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
1. lim (2x  1 
x 3
60
x (x  1)
2. xlim
0
2
lim2
x
 3x  1
3.
x 1
61
3
x
 2x
4. lim
x 2 5x  2
NOTE: : Direct substitution works in many
cases, so you should always try it first.
62
NOTE: These limit laws basically allow you to
do Direct Substitution.
x  2x
lim
4. x 2 5x  2
3
• Direct Substitution works in many cases, so
you should always try it first.
63
• However, there are a few cases
(mostly in math courses) where
direct substitution does not work
immediately, or at all.
64
y
A) Draw the graph of


x  2, x  3
f (x )  
2, x  3



x

B) Find lim f (x )
x 3















• 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
65
is at that x value.
Examples
1.
lim x
x 1
Direct substitution gives 1 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.
 lim x  DNE (Does Not Exist 
x 1
66
2.
lim
1
Examples
x2
1
Direct substitution gives
which
0
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:
x 0
lim
x 0
1
x
2

67
3.
x2 4
22  4 0


lim
x 2 x  2
22 0
Direct Substitution
0
0 , this means
• Whenever you get
there is some simplification you can do
to the function before you do the
direct substitution.
What would you do here??
Factor
x2 4
(x  2)(x  2)
lim
 lim
x 2 x  2
x 2
x 2
lim x  2  2  2  4
x 2
68
0
12  1  2
x2 x 2


4. lim
2
0
x 1 2x 2  x  1
2(1)  1  1 Direct
Substitution
What would you do here??
Factor
69
2h
(
lim
2
5. h 0
h
4
4  0
0
0
Direct Substitution
2  0
(

2
What would you do here??
More work!!
70
x 2 2
6. lim
x 2
x 2
22 4  0

0
22
Direct Substitution
Rationalize the
Numerator
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
71
x 2 2
lim
x 2
x 2
72
1 1 1
1
1

(2  0  
0
7. (2  h  
2 2 2 
2

lim
0
0
0
h 0
h
1
What would you do here??
Simplify the
rational expression
73
8. Find
(
lim
x 4
x  2( x  2 x  4 
x 4
74
9. Find
x x x 3 x 3
lim
x 1
x 1
75
10. Find
6 
 3x
lim 

x 2  x  2
x  2 
76
Practice:
x 2  2x  3
A) xlim
2
3 x  x  6
77
x3 1
B) lim 2
x 1 x  1
Practice:
78
Practice:
C)
9x
lim
x 9 3  x
79
Practice:
D) lim x  13  4
x 3
7 x 2
80
Page 44-45
# 3, 11, 14, 15, 17-19,21-28,
44,45
81
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
f (x )
g (x )
• 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
• 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 f (x )  f (b )
x b
This can be broken into 3 parts
1. f(b) is defined (It exists)
• b is in the domain of f(x)
2. lim f (x ) exists.
• In other words lim f
x b
x b
3. Part 1 = Part 2
(x )  lim f (x )
lim f (x )  f (b )
x b
x b
Describe why each place was
discontinuous

















Discuss the continuity of the
following
1. f(x) = x3 + 2x + 1
• This is continuous everywhere because it
is a polynomial.
2. g (x ) 
x
x 1
• Discontinuous at x = 1 (VA)
• 1 is not in the Domain
x  4, x  3
3.h (x )  
2, x  3
• Not continuous at x = 3. WHY?
lim f (x )  3  4  7  f (3)  2
x 3
1
 x , x  1
 2
4.f (x )  x , 1  x  1
x  1, x  1


• We need to check x = -1 and x = 1.
• Do we need to check x = 0?
x = -1
– NO! In 1/x, x=0 is not in x < -1
lim  f (x )  lim 
x 1
x 1
1
x
 1
lim  f (x )  lim   x 2  1
x 1
x 1
• Thus f(x) is continuous at x = 1
1
 x , x  1
 2
4.f (x )  x , 1  x  1
x  1, x  1


x=1
lim f (x )  lim  x
x 1
x 1
2
 1
lim f (x )  lim x  1  2
x 1
x 1
• Thus f(x) is discontinuous at x = 1 since
the left and right limits are not the same.
5. y = sinx
• Continuous everywhere
6. y = cos x
• Not continuous at VA
7. y = 2x
• Continuous everywhere
x 

2
 k ,k 
Examples
What value of k would make the
following functions continuous?
x 2  4
, x  2

1. f (x )   x  2
k , x  2


x  2x , x  2
2.h (x )  

5x  k , x  2
2
x  kx , x  1
3.f (x )  
kx  3, x  1
2
4. For what value of the constant c is the function
x  c , x  2
f (x )   2
cx  1, x  2
continuous at every number?
• 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
1
y  sin  
x 
y


• This is known
as an Oscillating
Discontinuity




x
















• 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.