Post on 28-Dec-2015
AP CALCULUS AB
Chapter 2:Limits and Continuity
Section 2.1:Rates of Change and Limits
What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem
…and why
Limits can be used to describe continuity, the derivative and the integral: the ideas giving the foundation of calculus.
Average and Instantaneous Speed
A body's average speed during an interval of time is
found by dividing the distance covered by the elapsed
time.
Experiments show that a dense solid object dropped
from rest to fall freely near the sur2
face of the earth will fall
16 feet in the first seconds. y t t
Average speed y
t
Section 2.1 – Rates of Change and Limits Instantaneous Speed:
To find the instantaneous speed, we start by calculating the average speed over an interval from time t1 to any slight time later t2=t1+h
To find the instantaneous speed, we take increasingly smaller values of h --- in other words, we let h approach 0.Then
h
tyhty
tht
tyhty
t
y 11
11
11
h
tfhtfh
0lim Speed ousInstantane
Definition of Limit
Let and be real numbers. The function has limit as
approaches if, given any positive number , there is a positive
number such that for all ,
0 .
We write
limx c
c L f L x
c
x
x c f x L
f x L
(If the horizontal distance between x and c is less than , then the vertical distance between and L is less than ).
or as x gets increasingly closer to c, then gets increasingly closer to L.
xf
xf
Definition of Limit continued
The sentence lim is read, "The limit of of as
approaches equals ". the notation means that the values
of the function approach or equal as the values of approach
(but do not equ
x cf x L f x x
c L f x
f L x
al) .
Figure 2.2 illustrates the fact that the existence of a limit as never
depends on how the function may or may not be defined at .
c
x c
c
Definition of Limit continued
The function has limit 2 as 1 even though is not defined at 1.
The function has limit 2 as 1 even though 1 2.
The function is the only one whose limit as 1 equals its value at =1.
f x f
g x g
h x x
Properties of Limits
If , , , and are real numbers and
lim and lim , then
1. : lim
The limit of the sum of two functions is the sum of their limits.
2. : lim
The limit
x c x c
x c
x c
L M c k
f x L g x M
Sum Rule f x g x L M
DifferenceRule f x g x L M
of the difference of two functions is the difference
of their limits.
Properties of Limits continued
3. lim
The limit of the product of two functions is the product of their limits.
4. lim
The limit of a constant times a function is the constant times the limit
of the function.
5.
x c
x c
f x g x L M
k f x k L
Quot
: lim , 0
The limit of the quotient of two functions is the quotient
of their limits, provided the limit of the denominator is not zero.
x c
f x Lient Rule M
g x M
Product Rule:
Constant Multiple Rule:
Properties of Limits continued
6. : If and are integers, 0, then
lim
provided that is a real number.
The limit of a rational power of a function is that power of the
limit of the function, provided the latte
rrss
x c
r
s
Power Rule r s s
f x L
L
r is a real number.
Other properties of limits:
lim
limx c
x c
k k
x c
Example Properties of
Limits
3
Use any of the properties of limits to find
lim 3 2 9x c
x x
3 3
3
sum and difference rules
product and multiple rules
lim 3 2 9 lim3 lim2 lim9
3 2 9
x c x c x c x cx x x x
c c
Polynomial and Rational Functions
11 0
11 0
1. If ... is any polynomial function and
is any real number, then
lim ...
n nn n
n nn nx c
f x a x a x a
c
f x f c a c a c a
2
4
2
Ex: 3 2 1 for
lim 3 2 1
3 16 2 4 1
48 8
4
1
5
4 4 4
5
x
f x x x c
f x f
Polynomial and Rational Functions
2. If and are polynomials and is any real number, then
lim , provided that 0.x c
f x g x c
f x f cg c
g x g c
2
3
Ex: 2 3 1 and for
2 3 1
3
3 9 3
3
10lim 10
123
2
x
g x x
g
f x x x c
f
g
x f
x
Example Limits 2
5Use Theorem 2 to find lim 4 2 6
xx x
22
5lim 4 -2 6 4 2 6 4 25 15 0 6 100 0 95 1 6 6x
x x
Section 2.1 – Rates of Change and Limits Techniques for Finding Limits
1. Numerically – plug in values that approach c from both the right and left.
2. Algebraically – factor and cancel/simplify first. Then plug in c for x.
3. Graphically.
In “well-behaved” functions we can find the by direct substitution of c for x:
xfcx
lim
3
4 3 4 7Ex: lim 7
2 3 2 1x
x
x
xfcx
lim
Evaluating LimitsAs with polynomials, limits of many familiar
functions can be found by substitution at points where they are defined. This includes trigonometric functions, exponential and logarithmic functions, and composites of these functions.
Section 2.1 – Rates of Change and Limits Limits of Trigonometric Functions:1. 2. 3. 4. 5. 6.
cx
cx
cx
cx
cx
cx
cx
cx
cx
cx
cx
cx
cotcotlim
secseclim
csccsclim
tantanlim
coscoslim
sinsinlim
Example Limits
Solve graphically:
1 sinThe graph of suggests that the limit exists and is 1.
cos
xf x
x
0
1 sinFind lim
cosx
x
x
0
00
Confirm Analytically:
lim 1 sin 1 sin 01 sinFind lim
cos lim cos cos0
1 0 1
1
x
xx
xx
x x
Example Limits0
5Find lim
x x
[-6,6] by [-10,10]
5Solve graphically: The graph of suggests that
the limit does not exist.
f xx
Confirm Analytically :
We can't use substitution in this example because when is relaced by 0,
the denominator becomes 0 and the function is undefined.
This would suggest that we rely on the graph to s
x
ee that the
limit does not exist.
Section 2.1 – Rates of Change and Limits Functions that agree in all but 1 point
.limlim
and exists also
lim then exists, lim and , containing
intervalopen an in allfor If
xgxf
xfxgc
cxxgxf
cxcx
cxcx
Section 2.1 – Rates of Change and Limits Cancellation Techniques for finding Limits
If you have a rational function
5
312
32lim
1
132lim
1
32lim :Ex
on.substituti use then , offactor
a with cancel to then try ,0
0 so
0 and 0 and such that
1
1
2
1
x
x
xx
x
xx
xp
xqcq
cpcr
cqcpxq
xpxrxr
x
xx
Section 2.1 – Rates of Change and Limits Rationalization Techniques for Finding Limits
22
1202
122
1lim
22lim
22
22lim
22
2222lim
22lim
0
0
0
0
0
x
xx
x
xx
xx
x
x
x
x
x
x
x
x
x
x
Section 2.1 – Rates of Change and Limits Special Limits from Trigonometry1.
2.
1sin
lim0
x
xx
0cos1
lim0
x
xx
Section 2.1 – Rates of Change and Limits Remember:
2233
2233
babababa
babababa
Section 2.1 – Rates of Change and Limits
As you approach 2 from the left, you get closer and closer to 12.
As you approach 2 from the right, you get closer and closer to 12.
So,
2
422
2
8)(
2
3
x
xxx
x
xxf
122
8lim
3
2
x
xx
Section 2.1 – Rates of Change and Limits Limit of a Composite Function:
If and
then
Lxgcx
)(lim LfxfLx
)(lim
Lfxgfcx
)(lim
One-Sided and Two-Sided LimitsSometimes the values of a function tend to different limits as approaches a
number from opposite sides. When this happens, we call the limit of as
approaches from the right the right-ha
f x
c f x
c
nd limit of at and the limit as
approaches from the left the left-hand limit.
right-hand: lim The limit of as approaches from the right.
left-hand: lim The limit of as ap
x c
x c
f c x
c
f x f x c
f x f x
proaches from the left.c
One-Sided and Two-Sided Limits continued
We sometimes call lim the two-sided limit of at to distinguish it from
the one-sided right-hand and left-hand limits of at .
A function has a limit as approaches if and only if the r
x cf x f c
f c
f x x c
ight-hand
and left-hand limits at exist and are equal. In symbols,
lim lim and lim .x c x c x c
c
f x L f x L f x L
Section 2.1 – Rates of Change and LimitsIf
As x approaches 2 from the left, f(x) approaches 3.
As x approaches 2 from the right, f(x) approaches 3.
So,
2 ,1
2 ,3)2(2)(
x
xxxf
3)(lim2
xfx
Example One-Sided and Two-Sided Limits
o
1 2 3
4
Find the following limits from the given graph.
0
2
2
2
3
a. lim
b. lim
c. lim
d. lim
e. lim
x
x
x
x
x
f x
f x
f x
f x
f x
0
Does Not Exist
4
Does Not Exist
0
Section 2.1 – Rates of Change and Limits Limits that do not exist:
As x approaches 0 from the left, f(x) approaches 0.
As x approaches 0 from the right, f(x) approaches 1.
So,
does not exist
0 ,
0 ,1)(
3
xx
xxxf
)(lim0
xfx
Section 2.1 – Rates of Change and Limits More Limits that do not
exist:
As x approaches , from the left, f(x) goes to
As x approaches , from the right, f(x) goes to
So
does not exist.
xxf tan)(
2
2
)(lim2
xfx
Section 2.1 – Rates of Change and Limits More Limits that do not exist:
Oscillating behavior
Graph on calculator and zoom in about 4 times around x=0.
xxf
1sin)(
Sandwich Theorem
If we cannot find a limit directly, we may be able to find
it indirectly with the Sandwich Theorem. The theorem
refers to a function whose values are sandwiched between
the values of two other func
f
tions, and .
If and have the same limit as then has that limit too.
g h
g h x c f
Sandwich Theorem
If for all in some interval about , and
lim =lim = ,
then
lim =
x c x c
x c
g x f x h x x c c
g x h x L
f x L
Section 2.1 – Rates of Change and Limits The Sandwich Theorem:If for all In some interval about c, and
Then
)()()( xhxfxg cx
Lxhxgcxcx
)(lim)(lim
Lxfcx
)(lim
g(x)
h(x)
f(x)