Post on 28-Dec-2015
2.1
Rates of Change and Limits
Quick ReviewIn Exercises 1 – 4, find f (2).
452 .1 23 xxxf 4
54 .2
3
2
x
xxf
2 sin .3x
xf
2 ,1
1
2 ,13 .4
2x
x
xxxf
0
12
11
0
3
1
Quick ReviewIn Exercises 5 – 8, write the inequality in the form a < x < b.
4 .5 x 2 .6 cx
32 .7 x 2 .8 dcx
44 x 22 cxc
51 x 22 dcxdc
Quick ReviewIn Exercises 9 and 10, write the fraction in reduced form.
3
183 .9
2
x
xx12
2 .10
2
2
xx
xx 6x 3x
6x
x 12 x
12 x 1x
1x
x
What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem
Essential QuestionHow can limits be used to describe continuity, the derivative
and the integral: the ideas giving the foundation of
Calculus?
Average and Instantaneous SpeedA 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 surface of the earth will fall y = 16t 2 feet in the first t seconds.
1. Wile E Coyote drops an anvil from the top of a cliff. What is its average rate of speed during the first 5 seconds of its fall?
traveleddistance timeelapsed t
y
22 016516
05
80 ft/sec
Average and Instantaneous SpeedA 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 surface of the earth will fall y = 16t 2 feet in the first t seconds.
2. What is the speed of the anvil at the instant t = 5?
t
y
22 516516
h
55 h
Since we cannot calculate the speed right at 5 sec. Calculate the average between 5 sec and slightly after 5 sec.
2516102516 2
hhh
h
hh 216160 h16160
The smaller h becomes the closer the average will get to 160.
Definition of LimitLet c and L be real numbers. The function f has limit L as x approaches c if, given any positive number , there is a positive number such that for all x,
We write
.0 Lxfcx
Lxfcx
lim
The sentence is read, “The limit of f of x as x approaches
c equals L.” The notation means that the values of x approach (but does
not equal) c.
Lxfcx
lim
The next figures illustrate the fact that the existence of a limit as x → c never depend on how the function may not be defined at c.
Definition of Limit continuedThe function f has a limit 2 as x → 1 even though f is not defined at 1.
The function g has a limit 2 as x → 1 even though g(1) ≠ 2.
The function h is the only one whose limit as x → 1 equals its value at x = 1.
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 LimitsUse any properties of limits to find:
2135lim .3 4
xxcx
Sum and difference rules45lim xcx
xcx13lim
2lim
cx
Product and multiple rules45c 13c 2
8
4lim .4
2
3
x
xxcx
Sum and difference rules lim 3xcx
xcx4lim
2lim xcx
Product and multiple rules c3
4c
2c
Quotient rule 8lim
4lim2
3
x
xx
cx
cx
8limcx
8
Polynomial and Rational Functions
11 0
11 0
1. If ... is any polynomial function and
is any real number, then
lim ...
2. If and are polynomials and is any real number, then
lim , prov
n nn n
n nn nx c
x c
f x a x a x a
c
f x f c a c a c a
f x g x c
f x f c
g x g c
ided that 0.g c
Example Limits5. Determine the limit by substitution. Support graphically.
143lim 2
2
xx
x 23 124 2
90 6 15
Evaluating LimitsAs with polynomials, limits of many familiar functionscan be found by substitution at points where they aredefined. This includes trigonometric functions, exponential and logarithmic functions, and compositesof these functions.
You can evaluate limits either graphically, numerically, or algebraically.
Example Limits6. Determine the limit by substitution. Support graphically.
5
12lim
2
3
x
xxx
1332 2 53
8
16
2
Example Limits
x
xx cos
sin1lim Find .7
0
Solve graphically:
1. is and existslimit that thesuggests cos
sin1 ofgraph The
x
xxf
Confirm algebraically:
x
xx cos
sin1lim
0
xx
sin1lim0
xx
coslim0
0sin1
01
0cos 11
Determine the limit graphically. Support algebraically.
Example Limits
xx
5lim Find .8
0
Solve graphically:
exist.not doeslimit that thesuggests 5
ofgraph Thex
xf
Confirm algebraically: undefined. is 0
5 because possibleNot
Confirm numerically:
x
y
1.0 01.0 001.0 1.0 01.0 001.0
50 500 5000 50 500 5000
Limit does not exist.
Determine the limit graphically. Support algebraically.
Example Limits
25
158lim Find .9
2
2
5
x
xxx
Solve graphically:
.5
1 is and existslimit that thesuggests
25
158 ofgraph The
2
2
x
xxxf
Confirm algebraically:
25
158lim
2
2
5
x
xxx
35lim
5
xxx 55 xx
3lim
5
xx
2 5x 10 5
1
Determine the limit graphically. Support algebraically.
Example Limits
xxx
xx 2
3
0 cos
sinlim Find .10
Solve graphically:
1. is and existslimit that thesuggests cos
sin ofgraph The
2
3
xxx
xxf
Confirm algebraically:
xxx
xx 2
3
0 cos
sinlim
sinlim
3
0
xx
xx 2cos1 xx
xx 2
3
5 sin
sinlim
x
xx
sinlim
0
xsin
1
Determine the limit graphically. Support algebraically.
On page 60, the graph and table shows that this limit approaches 1.
One-Sided and Two-Sided Limits
Sometimes 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 apx c
x c
f c x
c
f x f x c
f x f x
proaches from the left.c
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
Example One-Sided and Two-Sided Limitsx
xintlim 11.
2
Solve graphically:
2
xx
intlim 12.1
Solve graphically:
0
Example One-Sided and Two-Sided LimitsGiven the following graph, compute the limits.
4 13. f
DNE
xfx 4lim 14.
2
xfx 4lim 15.
2
xfx 4lim 16.
2
Example One-Sided and Two-Sided LimitsGiven the following graph, compute the limits.
1 17. f
4
xfx 1lim 18.
4
xfx 1lim 19.
2
xfx 1lim 20.
DNE
Example One-Sided and Two-Sided LimitsGiven the following graph, compute the limits.
6 21. f
2
xfx 6lim 22.
5
xfx 6lim 23.
5
xfx 6lim 24.
5
Sandwich TheoremIf 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
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
Pg. 66, 2.1 #1-47 odd