200701 Umaths Unit2-drling
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WUC112 University Mathematics
Unit 2: Limits and Continuity Tutor: Dr. Ling L. P.
Ground Rules Learn! Participate - ask questions. Take notes! Willing to share! Help each other learn - if you get it and your neighbor does not, help them. Help us to help you - let us know if we are going too fast or too slow. Be on time. 100% course work completion.
Dr. Ling L. P. 2007
Contact Information 1) Contact number: 04-530 8611
2) Email address: [email protected] / [email protected] This will be the best way to contact me!
3) Time available for telephone tutoring: Tuesday: 5.00 pm – 7.00 pm Thursday: 5.00 pm – 7.00 pm Please contact me at least once every 3 weeks.
Dr. Ling L. P. 2007
Unit 2 Objectives • • • • • • • •
Understand the idea of the limit of a function. Determine the existence of a limit of a point by looking at the right-hand limit and the left-hand limit of the point. Apply the limit theorems on the sum, product and quotient of functions. Understand the concept of continuity at a point of a particular function. Apply the theorems on continuity. Understand the idea of continuity from the left and continuity from the right. Apply the pinching theorem on some simple trigonometric limits. Understand and apply the intermediate value theorem and the extreme value theorem.
Dr. Ling L. P. 2007
Unit 2 Outline Unit 2.1: The idea of limits Unit 2.2: The definition of limits Unit 2.3: Some limit theorems Unit 2.4: Continuity
Dr. Ling L. P. 2007
Unit 2.1: The idea of limits
Concept of a Limit
Michael
??? 1 2
3 4
7 8
…
If Michael walks half way of the remaining distance each time, can he ever walk out of the room? If the concept of limit doesn’t exist, Michael can never walk out of the room! Dr. Ling L. P. 2007
Unit 2.2: The definition of limits
Function and Limit Function y = f ( x ) Examples of functions defined on all real numbers R. f(x) = x²; g(x) = 2x - 1; h(x) = x³
Definition: The limit of a function describes the behavior of the function as the input approaches a particular value.
f(x) = L lim x c Dr. Ling L. P. 2007
Think of the bus jumping the gap in “Speed” If the two pieces of the freeway are the same height...
…so, the jump works in either direction! Therefore, the limit exists.
Dr. Ling L. P. 2007
Think of the bus jumping the gap in “Speed” The left piece is higher…
…so, the jump works! Dr. Ling L. P. 2007
Think of the bus jumping the gap in “Speed” The right piece is lower…
…so, the jump fails!
Dr. Ling L. P. 2007
LIMIT does not exist (abbreviated D.N.E.)
Because the bus cannot jump both directions, then the limit of the freeway does not exist (abbreviated D.N.E.)
Dr. Ling L. P. 2007
One-sided limits You may find the limit from each side of the c-value. 12 11
x=c
Limit from the left (x
10 9
Limit from the right (x>c)
8 7 6 5 4 3 2 1 0 -6
-5
-4
-3
-2
-1
-1 0
1
2
3
4
5
6
-2 -3 -4
The hole at x = c divides the graph into two pieces! Dr. Ling L. P. 2007
One-sided limits
You need one-sided limits when you encounter a piecewise function (a function where each part of the domain has its own function to evaluate) as seen below… Limit from the left (x
Dr. Ling L. P. 2007
Limit from the right (x>c)
Example: Different right-hand and left-hand limits at the origin.
Dr. Ling L. P. 2007
Unit 2.3: Some limit theorems
Limit theorems
Dr. Ling L. P. 2007
Unit 2.4: Continuity
Definition of Continuity Definition: A function f is said to be continuous at a point a if and only if the limit of f as x approaches a is the value f (a) of f at a. That is:
lim f ( x) = f (a ) x →a
Dr. Ling L. P. 2007
Definition of Left Continuity Definition: A function f is said to be left continuous at a point a if and only if the limit of f as x approaches a from the left is the value f (a) of f at a.
lim f ( x) = f (a )
x →a −
Dr. Ling L. P. 2007
Definition of Right Continuity
Definition: A function f is said to be right continuous at a point a if and only if the limit of f as x approaches a from the right is the value f (a) of f at a.
lim f ( x) = f (a )
x →a +
Dr. Ling L. P. 2007
Continuity theorems
Dr. Ling L. P. 2007
Examples
Example 1
Solution
Dr. Ling L. P. 2007
( x + 2) lim 2 x →−2 x − 4
Problem 2
Solution
x 2 3x 2 lim x 2 x 2 x 2 3 x 2 x 1 x 2 Rewrite x 1. x 2 x 2 x 2 3x 2 Hence lim lim x 1 1. x 2 x 2 x 2
Dr. Ling L. P. 2007
Problem 3
Solution
x − 3x − 18 lim x→−3 x+3 2
( x + 3)( x − 6) lim x→−3 x+3 lim ( x − 6) x→−3
−3 − 6 −9 Dr. Ling L. P. 2007
Example 4
x
lim
4
x– 2 = ? x–4
Direct substitution won’t do it. Why?
Dr. Ling L. P. 2007
Solution
x– 2 •
x–4
Dr. Ling L. P. 2007
x+ 2
=
x– 4
x + 2 (x – 4)( x + 2)
Thus,
x
Dr. Ling L. P. 2007
lim
4
1 x+2
=?
Problem 5
Solution
Rewrite
x 0
6x
Use the fact that lim
0
sin 3 x
Since lim
x 0
Dr. Ling L. P. 2007
lim
sin 3 x
6x
1 sin 3 x 2 3x
sin 3 x 3x
sin
1.
1, we conclude that lim
x 0
sin 3 x 6x
1 . 2
Practice: 1)
x
5)
x
lim
1
lim
Dr. Ling L. P. 2007
–
x –4 =? 2
x+2 x –4 =? 2
2
By direct substitution
x+2
By factoring and substituting
Example 6
Determine if the limit exists as the function approaches 3?
f (x) =
x+1
if x ≤ 3
8 – 2x
if x > 3
Solution
For the “left” or negative approach, try to evaluate the first piece
4
lim (x + 1) = This is NOT – 3
Dr. Ling L. P. 2007
x3–
“approaching 3 negatively”
For the “right” or positive approach, try to evaluate the second piece
2
lim (8 – 2x) = This is STILL 3
x3+
What is the limit as this piecewise function approaches 3? 4 from the left and 2 from the right Thus,
x
lim f(x)
Dr. Ling L. P. 2007
3
does not exist (D.N.E.)
Limit of function when approaches infinity Example 7
Find the limit as the function approaches 2.
f (x) =
1 2− x
Tips
Solution
Plot it!! In the graph, as x 2– , the curve + ∞ In the graph, as x 2+ , the curve – ∞ The two one-sided limits do not agree; Thus, limit D.N.E.
Dr. Ling L. P. 2007
More Examples Limits-Problems&Sol ution
Supporting data/information
Dr. Ling L. P. 2007
sin x f ( x) = x The function is undefined at 0. Table of Values
sin x lim =1 x→0 x Dr. Ling L. P. 2007
x
sin(x)/x
±0.5
0.95885
±0.1
0.99833
±0.001
0.99999
Graph for example 7
x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Dr. Ling L. P. 2007
f(x) 0.083333 0.090909 0.1 0.111111 0.125 0.142857 0.166667 0.2 0.25 0.333333 0.5 1 -1 -0.5 -0.333333 -0.25 -0.2 -0.166667 -0.142857 -0.125
Unit 2: Limits and Continuity Tutor: Dr. Ling L. P.
Ground Rules Learn! Participate - ask questions. Take notes! Willing to share! Help each other learn - if you get it and your neighbor does not, help them. Help us to help you - let us know if we are going too fast or too slow. Be on time. 100% course work completion.
Dr. Ling L. P. 2007
Contact Information 1) Contact number: 04-530 8611
2) Email address: [email protected] / [email protected] This will be the best way to contact me!
3) Time available for telephone tutoring: Tuesday: 5.00 pm – 7.00 pm Thursday: 5.00 pm – 7.00 pm Please contact me at least once every 3 weeks.
Dr. Ling L. P. 2007
Unit 2 Objectives • • • • • • • •
Understand the idea of the limit of a function. Determine the existence of a limit of a point by looking at the right-hand limit and the left-hand limit of the point. Apply the limit theorems on the sum, product and quotient of functions. Understand the concept of continuity at a point of a particular function. Apply the theorems on continuity. Understand the idea of continuity from the left and continuity from the right. Apply the pinching theorem on some simple trigonometric limits. Understand and apply the intermediate value theorem and the extreme value theorem.
Dr. Ling L. P. 2007
Unit 2 Outline Unit 2.1: The idea of limits Unit 2.2: The definition of limits Unit 2.3: Some limit theorems Unit 2.4: Continuity
Dr. Ling L. P. 2007
Unit 2.1: The idea of limits
Concept of a Limit
Michael
??? 1 2
3 4
7 8
…
If Michael walks half way of the remaining distance each time, can he ever walk out of the room? If the concept of limit doesn’t exist, Michael can never walk out of the room! Dr. Ling L. P. 2007
Unit 2.2: The definition of limits
Function and Limit Function y = f ( x ) Examples of functions defined on all real numbers R. f(x) = x²; g(x) = 2x - 1; h(x) = x³
Definition: The limit of a function describes the behavior of the function as the input approaches a particular value.
f(x) = L lim x c Dr. Ling L. P. 2007
Think of the bus jumping the gap in “Speed” If the two pieces of the freeway are the same height...
…so, the jump works in either direction! Therefore, the limit exists.
Dr. Ling L. P. 2007
Think of the bus jumping the gap in “Speed” The left piece is higher…
…so, the jump works! Dr. Ling L. P. 2007
Think of the bus jumping the gap in “Speed” The right piece is lower…
…so, the jump fails!
Dr. Ling L. P. 2007
LIMIT does not exist (abbreviated D.N.E.)
Because the bus cannot jump both directions, then the limit of the freeway does not exist (abbreviated D.N.E.)
Dr. Ling L. P. 2007
One-sided limits You may find the limit from each side of the c-value. 12 11
x=c
Limit from the left (x
10 9
Limit from the right (x>c)
8 7 6 5 4 3 2 1 0 -6
-5
-4
-3
-2
-1
-1 0
1
2
3
4
5
6
-2 -3 -4
The hole at x = c divides the graph into two pieces! Dr. Ling L. P. 2007
One-sided limits
You need one-sided limits when you encounter a piecewise function (a function where each part of the domain has its own function to evaluate) as seen below… Limit from the left (x
Dr. Ling L. P. 2007
Limit from the right (x>c)
Example: Different right-hand and left-hand limits at the origin.
Dr. Ling L. P. 2007
Unit 2.3: Some limit theorems
Limit theorems
Dr. Ling L. P. 2007
Unit 2.4: Continuity
Definition of Continuity Definition: A function f is said to be continuous at a point a if and only if the limit of f as x approaches a is the value f (a) of f at a. That is:
lim f ( x) = f (a ) x →a
Dr. Ling L. P. 2007
Definition of Left Continuity Definition: A function f is said to be left continuous at a point a if and only if the limit of f as x approaches a from the left is the value f (a) of f at a.
lim f ( x) = f (a )
x →a −
Dr. Ling L. P. 2007
Definition of Right Continuity
Definition: A function f is said to be right continuous at a point a if and only if the limit of f as x approaches a from the right is the value f (a) of f at a.
lim f ( x) = f (a )
x →a +
Dr. Ling L. P. 2007
Continuity theorems
Dr. Ling L. P. 2007
Examples
Example 1
Solution
Dr. Ling L. P. 2007
( x + 2) lim 2 x →−2 x − 4
Problem 2
Solution
x 2 3x 2 lim x 2 x 2 x 2 3 x 2 x 1 x 2 Rewrite x 1. x 2 x 2 x 2 3x 2 Hence lim lim x 1 1. x 2 x 2 x 2
Dr. Ling L. P. 2007
Problem 3
Solution
x − 3x − 18 lim x→−3 x+3 2
( x + 3)( x − 6) lim x→−3 x+3 lim ( x − 6) x→−3
−3 − 6 −9 Dr. Ling L. P. 2007
Example 4
x
lim
4
x– 2 = ? x–4
Direct substitution won’t do it. Why?
Dr. Ling L. P. 2007
Solution
x– 2 •
x–4
Dr. Ling L. P. 2007
x+ 2
=
x– 4
x + 2 (x – 4)( x + 2)
Thus,
x
Dr. Ling L. P. 2007
lim
4
1 x+2
=?
Problem 5
Solution
Rewrite
x 0
6x
Use the fact that lim
0
sin 3 x
Since lim
x 0
Dr. Ling L. P. 2007
lim
sin 3 x
6x
1 sin 3 x 2 3x
sin 3 x 3x
sin
1.
1, we conclude that lim
x 0
sin 3 x 6x
1 . 2
Practice: 1)
x
5)
x
lim
1
lim
Dr. Ling L. P. 2007
–
x –4 =? 2
x+2 x –4 =? 2
2
By direct substitution
x+2
By factoring and substituting
Example 6
Determine if the limit exists as the function approaches 3?
f (x) =
x+1
if x ≤ 3
8 – 2x
if x > 3
Solution
For the “left” or negative approach, try to evaluate the first piece
4
lim (x + 1) = This is NOT – 3
Dr. Ling L. P. 2007
x3–
“approaching 3 negatively”
For the “right” or positive approach, try to evaluate the second piece
2
lim (8 – 2x) = This is STILL 3
x3+
What is the limit as this piecewise function approaches 3? 4 from the left and 2 from the right Thus,
x
lim f(x)
Dr. Ling L. P. 2007
3
does not exist (D.N.E.)
Limit of function when approaches infinity Example 7
Find the limit as the function approaches 2.
f (x) =
1 2− x
Tips
Solution
Plot it!! In the graph, as x 2– , the curve + ∞ In the graph, as x 2+ , the curve – ∞ The two one-sided limits do not agree; Thus, limit D.N.E.
Dr. Ling L. P. 2007
More Examples Limits-Problems&Sol ution
Supporting data/information
Dr. Ling L. P. 2007
sin x f ( x) = x The function is undefined at 0. Table of Values
sin x lim =1 x→0 x Dr. Ling L. P. 2007
x
sin(x)/x
±0.5
0.95885
±0.1
0.99833
±0.001
0.99999
Graph for example 7
x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Dr. Ling L. P. 2007
f(x) 0.083333 0.090909 0.1 0.111111 0.125 0.142857 0.166667 0.2 0.25 0.333333 0.5 1 -1 -0.5 -0.333333 -0.25 -0.2 -0.166667 -0.142857 -0.125
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