View
215
Download
1
Category
Preview:
Citation preview
Math 114Calculus I
Instructor Christopher Davis
September 1, 2015
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 1 / 15
Who am I?
Christopher Davis,Office: 533 Hibbard Humanities Hall
eMail: daviscw@uwec.eduOffice hours: Monday, Tuesday,
Wednesday, Friday 3 - 4 PM(also by appointment,
but please give me warning)
I prefer the film Alien overthe film Predator.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 2 / 15
Who am I?
Christopher Davis,Office: 533 Hibbard Humanities Hall
eMail: daviscw@uwec.eduOffice hours: Monday, Tuesday,
Wednesday, Friday 3 - 4 PM(also by appointment,
but please give me warning)
I prefer the film Alien overthe film Predator.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 2 / 15
What is Calculus?
Any opinions?
At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.
For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?
p(1)−p(.5)1−.5 = (20−5)−(20−1.25)
.5 = −3.75.5 = −7.5
From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 =
(20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 =
−3.75.5 = −7.5
From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5
From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1?
−9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5
From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1?
−9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95
Trick question From t0 = 1 to t1 = 1? 00 makes no sense. The average
velocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1?
00 makes no sense. The average
velocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense.
The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.
Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?
Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to?
10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.
The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
What is Calculus?
Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)
1−.5 = (20−5)−(20−1.25).5 = −3.75
.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0
0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4 / 15
Limits
The core of calculus is the concept of the Limit. I will introduce it viaanother example.
Consider the function given by the for-mula
f (h) =1 − h2
1 − h
If you take h = 1 The formula becomes 00
which does not make any sense.Here is the graph of the function.
If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?Answer 2.
We say that the limit of f (h) as h approaches 1 is 2. As a shorthand wewrite lim
h→1f (h) = 2.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 5 / 15
Limits
The core of calculus is the concept of the Limit. I will introduce it viaanother example.Consider the function given by the for-mula
f (h) =1 − h2
1 − h
If you take h = 1 The formula becomes 00
which does not make any sense.Here is the graph of the function.
If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?Answer 2.
We say that the limit of f (h) as h approaches 1 is 2. As a shorthand wewrite lim
h→1f (h) = 2.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 5 / 15
Limits
The core of calculus is the concept of the Limit. I will introduce it viaanother example.Consider the function given by the for-mula
f (h) =1 − h2
1 − h
If you take h = 1 The formula becomes 00
which does not make any sense.
Here is the graph of the function.
If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?Answer 2.
We say that the limit of f (h) as h approaches 1 is 2. As a shorthand wewrite lim
h→1f (h) = 2.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 5 / 15
Limits
The core of calculus is the concept of the Limit. I will introduce it viaanother example.Consider the function given by the for-mula
f (h) =1 − h2
1 − h
If you take h = 1 The formula becomes 00
which does not make any sense.Here is the graph of the function.
If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?Answer 2.We say that the limit of f (h) as h approaches 1 is 2. As a shorthand wewrite lim
h→1f (h) = 2.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 5 / 15
Limits
The core of calculus is the concept of the Limit. I will introduce it viaanother example.Consider the function given by the for-mula
f (h) =1 − h2
1 − h
If you take h = 1 The formula becomes 00
which does not make any sense.Here is the graph of the function.If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?
Answer 2.We say that the limit of f (h) as h approaches 1 is 2. As a shorthand wewrite lim
h→1f (h) = 2.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 5 / 15
Limits
The core of calculus is the concept of the Limit. I will introduce it viaanother example.Consider the function given by the for-mula
f (h) =1 − h2
1 − h
If you take h = 1 The formula becomes 00
which does not make any sense.Here is the graph of the function.If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?Answer 2.
We say that the limit of f (h) as h approaches 1 is 2. As a shorthand wewrite lim
h→1f (h) = 2.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 5 / 15
Limits
The core of calculus is the concept of the Limit. I will introduce it viaanother example.Consider the function given by the for-mula
f (h) =1 − h2
1 − h
If you take h = 1 The formula becomes 00
which does not make any sense.Here is the graph of the function.If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?Answer 2.We say that the limit of f (h) as h approaches 1 is 2. As a shorthand wewrite lim
h→1f (h) = 2.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 5 / 15
Informal definition of limits.
Definition
We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.
Notice that the definition of the limit does not care what value thefunction takes at x = a.Example:
Here is the graph of f (x) = x3−8x−2 − 8.
When x is close to 2 what is f (x) closeto?
What is limx→2
x3 − 8
x − 2− 8?
limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 6 / 15
Informal definition of limits.
Definition
We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.
Notice that the definition of the limit does not care what value thefunction takes at x = a.
Example:
Here is the graph of f (x) = x3−8x−2 − 8.
When x is close to 2 what is f (x) closeto?
What is limx→2
x3 − 8
x − 2− 8?
limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 6 / 15
Informal definition of limits.
Definition
We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.
Notice that the definition of the limit does not care what value thefunction takes at x = a.Example:
Here is the graph of f (x) = x3−8x−2 − 8.
When x is close to 2 what is f (x) closeto?
What is limx→2
x3 − 8
x − 2− 8?
limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 6 / 15
Informal definition of limits.
Definition
We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.
Notice that the definition of the limit does not care what value thefunction takes at x = a.Example:
Here is the graph of f (x) = x3−8x−2 − 8.
When x is close to 2 what is f (x) closeto?
What is limx→2
x3 − 8
x − 2− 8?
limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 6 / 15
More limits from graphs: How close to x need to be to a?
(1,0) 1.9 2.1(1,0)
y=3
y=5Here is the graph of f (x) = x3−8
x−2 − 8, againjust zoomed in some.
If x is within .1 of 2 (between 1.9 and 2.1),then according to this graph how close do youknow that f (x) is to 4?Answer: It looks like 3 < f (x) < 5. f (x) iswithin 1 of 4.You can guarantee that f (x) is within 1 of 4by insisting that x be within .1 of 2.The moral: The graph of f (x) does morethan tell you about limits. It tells you howx has to be to a for f (x) to be close to thelimit.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 7 / 15
More limits from graphs: How close to x need to be to a?
(1,0) 1.9 2.1(1,0)
y=3
y=5Here is the graph of f (x) = x3−8
x−2 − 8, againjust zoomed in some.If x is within .1 of 2 (between 1.9 and 2.1),then according to this graph how close do youknow that f (x) is to 4?
Answer: It looks like 3 < f (x) < 5. f (x) iswithin 1 of 4.You can guarantee that f (x) is within 1 of 4by insisting that x be within .1 of 2.The moral: The graph of f (x) does morethan tell you about limits. It tells you howx has to be to a for f (x) to be close to thelimit.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 7 / 15
More limits from graphs: How close to x need to be to a?
(1,0) 1.9 2.1(1,0)
y=3
y=5Here is the graph of f (x) = x3−8
x−2 − 8, againjust zoomed in some.If x is within .1 of 2 (between 1.9 and 2.1),then according to this graph how close do youknow that f (x) is to 4?Answer: It looks like 3 < f (x) < 5. f (x) iswithin 1 of 4.
You can guarantee that f (x) is within 1 of 4by insisting that x be within .1 of 2.The moral: The graph of f (x) does morethan tell you about limits. It tells you howx has to be to a for f (x) to be close to thelimit.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 7 / 15
More limits from graphs: How close to x need to be to a?
(1,0) 1.9 2.1(1,0)
y=3
y=5Here is the graph of f (x) = x3−8
x−2 − 8, againjust zoomed in some.If x is within .1 of 2 (between 1.9 and 2.1),then according to this graph how close do youknow that f (x) is to 4?Answer: It looks like 3 < f (x) < 5. f (x) iswithin 1 of 4.You can guarantee that f (x) is within 1 of 4by insisting that x be within .1 of 2.
The moral: The graph of f (x) does morethan tell you about limits. It tells you howx has to be to a for f (x) to be close to thelimit.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 7 / 15
More limits from graphs: How close to x need to be to a?
(1,0) 1.9 2.1(1,0)
y=3
y=5Here is the graph of f (x) = x3−8
x−2 − 8, againjust zoomed in some.If x is within .1 of 2 (between 1.9 and 2.1),then according to this graph how close do youknow that f (x) is to 4?Answer: It looks like 3 < f (x) < 5. f (x) iswithin 1 of 4.You can guarantee that f (x) is within 1 of 4by insisting that x be within .1 of 2.The moral: The graph of f (x) does morethan tell you about limits. It tells you howx has to be to a for f (x) to be close to thelimit.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 7 / 15
x=.5 x=1.5
y=2
y=4
x=.75 x=1.25
y=2.5
y=3.5
Get a guess for the value of limx→1
2x + 1. (plotted above)
Using the picture above, how close to 1 does x need to be if you want2x + 1 to be within 1 of 3?Within .5 of 3?
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 8 / 15
x=.5 x=1.5
y=2
y=4
x=.75 x=1.25
y=2.5
y=3.5
Get a guess for the value of limx→1
2x + 1. (plotted above)
Using the picture above, how close to 1 does x need to be if you want2x + 1 to be within 1 of 3?
Within .5 of 3?
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 8 / 15
x=.5 x=1.5
y=2
y=4
x=.75 x=1.25
y=2.5
y=3.5
Get a guess for the value of limx→1
2x + 1. (plotted above)
Using the picture above, how close to 1 does x need to be if you want2x + 1 to be within 1 of 3?Within .5 of 3?
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 8 / 15
Getting guesses for limits from tables
Let’s determine the limit of f (x) = x3−8x−2 − 8 as x approaches 2 again.
Here is our strategy, we will take choices of x which get close to 2 (butdon’t equal 2) and evaluate f (x) at these inputs.
Let’s fill in this table.x 1 1.9 1.99 1.999 1.9999
x3−8x−2 − 8
-1 3.41 3.94 3.99 3.999
What number does it look like these values are approaching?
At this point we guess that limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 9 / 15
Getting guesses for limits from tables
Let’s determine the limit of f (x) = x3−8x−2 − 8 as x approaches 2 again.
Here is our strategy, we will take choices of x which get close to 2 (butdon’t equal 2) and evaluate f (x) at these inputs.Let’s fill in this table.
x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8
-1 3.41 3.94 3.99 3.999What number does it look like these values are approaching?
At this point we guess that limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 9 / 15
Getting guesses for limits from tables
Let’s determine the limit of f (x) = x3−8x−2 − 8 as x approaches 2 again.
Here is our strategy, we will take choices of x which get close to 2 (butdon’t equal 2) and evaluate f (x) at these inputs.Let’s fill in this table.
x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1
3.41 3.94 3.99 3.999What number does it look like these values are approaching?
At this point we guess that limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 9 / 15
Getting guesses for limits from tables
Let’s determine the limit of f (x) = x3−8x−2 − 8 as x approaches 2 again.
Here is our strategy, we will take choices of x which get close to 2 (butdon’t equal 2) and evaluate f (x) at these inputs.Let’s fill in this table.
x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41
3.94 3.99 3.999What number does it look like these values are approaching?
At this point we guess that limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 9 / 15
Getting guesses for limits from tables
Let’s determine the limit of f (x) = x3−8x−2 − 8 as x approaches 2 again.
Here is our strategy, we will take choices of x which get close to 2 (butdon’t equal 2) and evaluate f (x) at these inputs.Let’s fill in this table.
x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41 3.94
3.99 3.999What number does it look like these values are approaching?
At this point we guess that limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 9 / 15
Getting guesses for limits from tables
Let’s determine the limit of f (x) = x3−8x−2 − 8 as x approaches 2 again.
Here is our strategy, we will take choices of x which get close to 2 (butdon’t equal 2) and evaluate f (x) at these inputs.Let’s fill in this table.
x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41 3.94 3.99
3.999What number does it look like these values are approaching?
At this point we guess that limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 9 / 15
Getting guesses for limits from tables
Let’s determine the limit of f (x) = x3−8x−2 − 8 as x approaches 2 again.
Here is our strategy, we will take choices of x which get close to 2 (butdon’t equal 2) and evaluate f (x) at these inputs.Let’s fill in this table.
x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41 3.94 3.99 3.999
What number does it look like these values are approaching?
At this point we guess that limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 9 / 15
Getting guesses for limits from tables
Let’s determine the limit of f (x) = x3−8x−2 − 8 as x approaches 2 again.
Here is our strategy, we will take choices of x which get close to 2 (butdon’t equal 2) and evaluate f (x) at these inputs.Let’s fill in this table.
x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41 3.94 3.99 3.999
What number does it look like these values are approaching?
At this point we guess that limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 9 / 15
Getting guesses for limits from tables
Let’s determine the limit of f (x) = x3−8x−2 − 8 as x approaches 2 again.
Here is our strategy, we will take choices of x which get close to 2 (butdon’t equal 2) and evaluate f (x) at these inputs.Let’s fill in this table.
x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41 3.94 3.99 3.999
What number does it look like these values are approaching?
At this point we guess that limx→2
x3 − 8
x − 2− 8 = 4.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 9 / 15
For You:
Pick some sample points getting close to 1 and see what 2x + 1 gets closeto for your sample points.What do you predict that limx→1 2x + 1 is?
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 10 / 15
Building more limit intuition (a preview of limit laws.)
Ready to think in the abstract?
Definition
We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.
Last page we guessed that limx→1
2x + 1 = 3.
Let’s try to justify this.
Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get? Anumber really close to 22x is really close to 2What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 11 / 15
Building more limit intuition (a preview of limit laws.)
Ready to think in the abstract?
Definition
We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.
Last page we guessed that limx→1
2x + 1 = 3.
Let’s try to justify this.Suppose that x is really close to 1.
If we double a number that’s really close to 1 then what do we get? Anumber really close to 22x is really close to 2What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 11 / 15
Building more limit intuition (a preview of limit laws.)
Ready to think in the abstract?
Definition
We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.
Last page we guessed that limx→1
2x + 1 = 3.
Let’s try to justify this.Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get?
Anumber really close to 22x is really close to 2What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 11 / 15
Building more limit intuition (a preview of limit laws.)
Ready to think in the abstract?
Definition
We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.
Last page we guessed that limx→1
2x + 1 = 3.
Let’s try to justify this.Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get? Anumber really close to 2
2x is really close to 2What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 11 / 15
Building more limit intuition (a preview of limit laws.)
Ready to think in the abstract?
Definition
We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.
Last page we guessed that limx→1
2x + 1 = 3.
Let’s try to justify this.Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get? Anumber really close to 22x is really close to 2
What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 11 / 15
Building more limit intuition (a preview of limit laws.)
Ready to think in the abstract?
Definition
We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.
Last page we guessed that limx→1
2x + 1 = 3.
Let’s try to justify this.Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get? Anumber really close to 22x is really close to 2What happens if we add 1 to a number really close to 2?
2x + 1 is really close to 3.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 11 / 15
Building more limit intuition (a preview of limit laws.)
Ready to think in the abstract?
Definition
We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.
Last page we guessed that limx→1
2x + 1 = 3.
Let’s try to justify this.Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get? Anumber really close to 22x is really close to 2What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 11 / 15
Compute one more limit in this intuitive way.
Let’s compute limx→2 2x2 + 3x − 1.Let’s let x be really close to 2Complete these questions.
1 If you square a number (x) close to 2 what do you get?
2 If you double a number close to (The result of 1) then what doyou get?
3 If you multiply a number close to 2 by 3 what do you get?
4 If you start with a number close to (The result of 2), add(The result of 3) and then subtract 1 what do you get?
What is limx→2 2x2 + 3x − 1?
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 12 / 15
A function without a limit
Here is the graph of the sign function:There are typo’s on this page of notes.
sign(x) =
−1 x < 00 x = 0+1 x > 0
No matter how close to x = 0 you look, sign(x) is+1 and −1 in that neighborhood.
There is no number which is super close to each of −1 and +1.There is no limit.If the function approaches different things from the left and the right thenthere is no limit.The moral: If the graph of a function jumps, then the function has nolimit (Later: one-sided-limits)
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 13 / 15
A function without a limit
Here is the graph of the sign function:There are typo’s on this page of notes.
sign(x) =
−1 x < 00 x = 0+1 x > 0
No matter how close to x = 0 you look, sign(x) is+1 and −1 in that neighborhood.
There is no number which is super close to each of −1 and +1.
There is no limit.If the function approaches different things from the left and the right thenthere is no limit.The moral: If the graph of a function jumps, then the function has nolimit (Later: one-sided-limits)
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 13 / 15
A function without a limit
Here is the graph of the sign function:There are typo’s on this page of notes.
sign(x) =
−1 x < 00 x = 0+1 x > 0
No matter how close to x = 0 you look, sign(x) is+1 and −1 in that neighborhood.
There is no number which is super close to each of −1 and +1.There is no limit.
If the function approaches different things from the left and the right thenthere is no limit.The moral: If the graph of a function jumps, then the function has nolimit (Later: one-sided-limits)
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 13 / 15
A function without a limit
Here is the graph of the sign function:There are typo’s on this page of notes.
sign(x) =
−1 x < 00 x = 0+1 x > 0
No matter how close to x = 0 you look, sign(x) is+1 and −1 in that neighborhood.
There is no number which is super close to each of −1 and +1.There is no limit.If the function approaches different things from the left and the right thenthere is no limit.
The moral: If the graph of a function jumps, then the function has nolimit (Later: one-sided-limits)
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 13 / 15
A function without a limit
Here is the graph of the sign function:There are typo’s on this page of notes.
sign(x) =
−1 x < 00 x = 0+1 x > 0
No matter how close to x = 0 you look, sign(x) is+1 and −1 in that neighborhood.
There is no number which is super close to each of −1 and +1.There is no limit.If the function approaches different things from the left and the right thenthere is no limit.The moral: If the graph of a function jumps, then the function has nolimit (Later: one-sided-limits)
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 13 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x)
1 1 1 1What does it look like limx→0 cos(π/x) is?
This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x)
-1 -1 -1 -1
This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1
1 1 1What does it look like limx→0 cos(π/x) is?
This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x)
-1 -1 -1 -1
This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1
1 1What does it look like limx→0 cos(π/x) is?
This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x)
-1 -1 -1 -1
This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1
1What does it look like limx→0 cos(π/x) is?
This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x)
-1 -1 -1 -1
This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1
What does it look like limx→0 cos(π/x) is?
This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x)
-1 -1 -1 -1
This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?
This table predicts that it is 1x 1/11 1/101 1/1001 1/10001
cos(π/x)
-1 -1 -1 -1
This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x)
-1 -1 -1 -1
This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x)
-1 -1 -1 -1This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x) -1
-1 -1 -1This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x) -1 -1
-1 -1This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x) -1 -1 -1
-1This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x) -1 -1 -1 -1
This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x) -1 -1 -1 -1This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x) -1 -1 -1 -1This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).
It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x) -1 -1 -1 -1This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).It wiggles too much
super wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
A terrible function
Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board
x 1/10 1/100 1/1000 1/10000
cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?This table predicts that it is 1
x 1/11 1/101 1/1001 1/10001
cos(π/x) -1 -1 -1 -1This table thinks that the limit is −1.What gives? The limit can’t be both.
Here’s a graph of cos(π/x).It wiggles too muchsuper wiggly functions don’t have limits!
Instructor Christopher Davis Math 114 Calculus I September 1, 2015 14 / 15
Recommended