chapter 0ne limits and rates of change up down return end
TRANSCRIPT
Chapter 0ne
Limits and Rates of Change
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1.4 The Precise Definition of a Limit
?)(lim Lxfax
We know that it means f(x) is moving close to L while x is moving close to a as we desire. And it can reaches L as near as we like only on condition of the x is in a neighbor.
(2) DEFINITION Let f(x) be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write , if for very number >0 there is a corresponding number >0 such that
|f(x) - L|< whenever 0<|x - a|< .
Lxfax
)(lim
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How to give mathematical description of
In the definition, the main part is that for arbitrarily >0, there exists a >0 such that if all x that 0<|x - a|< then |f(x) - L|< .
Another notation for is f(x) L as x a.Lxfax
)(lim
Geometric interpretation of limits can be given in terms of the graph of the function
y=L+
y=L -
y=L
a a- a+o x
y=f(x) y
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Example 1 Prove that .7)54(lim3
xx
Solution Let be a given positive number, we want to find a positive number such that
|(4x-5)-7|< whenever 0<|x-3|<.
But |(4x-5)-7|=4|x-3|. Therefore
4|x-3|< whenever 0<|x-3|<.
That is, |x-3|< /4 whenever 0<|x-3|<.
Example 2 Prove that .8lim 3
2
x
x
Example 3 Prove that .sinsinlim axax
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Example 4 Prove that101
459
lim2
4
tt
t
Similarly we can give the definitions of one-sided limits precisely.
(4)DEFINITION OF LEFT-SIDED LIMIT
If for every number >0 there is a corresponding number >0 such that |f(x) - L|< whenever 0< a - x <, i.e, a - < x < a.
Lxfax
)(lim
(5)DEFINITION OF LEFT-SIDED LIMIT
If for every number >0 there is a corresponding number >0 such that |f(x) - L|< whenever 0< x - a <, i.e, a < x < a + .
Lxfax
)(lim
.0lim0
xx Example 5 Prove that
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Example 6 If Lxfax
)(lim ,)(lim, Mxgax
prove that ,)]()([lim MLxgxfax
,)]()([lim LMxgxfax
).0(,)()(
lim
MML
xgxf
ax
(6) DEFINITION Let f(x) be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is infinity, and we write , if for very number M>0 there is a corresponding number >0 such that
f(x)>M whenever 0< |x - a|< .
)(lim xfax
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.)1(
1lim 21
xx
Example Prove that
Example 5 Prove that
(6)DEFINITION Let f(x) be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is infinity, and we write , if for very number N<0 there is a corresponding number >0 such that
f(x)<N whenever 0< |x - a|< .
)(lim xfax
||lnlim0
xx
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Similarly, we can give the definitions of one-side infinite limits.
)(lim xfax
)(lim xfax
)(lim xfax
)(lim xfax
x
x
1
02limExample Prove that
Example Prove that 02lim1
0
x
x
11
lim1 xxExample Prove that
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1.5 Continuity
If f(x) not continuous at a, we say f(x) is discontinuous at a , or f(x) has a discontinuity at a .
(1) Definition A function f(x) is continuous at a number a if
)()(lim afxfax
. (3) )()(lim afxfax
A function f(x) is continuous at a number a if and only if for every number >0 there is a corresponding number >0 such that
|f(x) - f(a) |< whenever |x - a|< .
Note that: (1) f(a) is defined)(lim xf
ax(2) exists.
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Example is discontinuous at x=2, since f(2)
is not defined.
22
)(2
xxx
xf
Example is continuous at
x=2..
223
22
)(
2
x
xx
xxxf
Example Prove that sinx is continuous at x=a.
(2) Definition A function f(x) is continuous from the right at every
number a if
A function f(x) is continuous from the left at every
number a if
)()(lim afxfax
)()(lim afxfax
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(2) Definition A function f(x) is continuous on an interval if it is
continuous at every number in the interval. (at an endpoint of the interval we understand continuous to mean continuous from the right or continuous from the left)
Example At each integer n, the function f(x)=[x] is continuous from the right and discontinuous from the left.
Example Show that the function f(x)=1-(1-x2) 1/2 is continuous on the interval [-1,1].
(4)Theorem If functions f(x), g(x) is continuous at a and c is a constant, then the following functions are continuous at a:
1. f(x)+g(x) 2. f(x)-g(x) 3. f(x)g(x) 4. f(x)[g(x)] -1 (g(a) isn’t 0.)
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(5) THEOREM (a) any polynomial is continuous everywhere, that is, it is continuous on R1=().
(b) any rational function is continuous wherever it is defined, that is, it is continuous on its domain.
Example Find .25
53lim
2
2 xxx
x
(6) THEOREM If n is a positive even integer, then f(x)= is continuous on [0, ). If n is a positive odd integer, then f(x)= is continuous on ().
n xn x
Example On what intervals is each function continuous?
,4
53)( 2
2
xxx
a
.1
111
1)( 22
xx
xx
xb
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(8) THEOREM If g(x) is continuous at a and f(x) is continuous at g(a) then (fog)(x))= f(g(x)) is continuous at a .
(7) THE INTERMEDIATE VALUE THEOREM Suppose that f(x) is continuous on the closed interval [a,b]. Let N be any number strictly between f(a) and f(b). Then there exists a number c in (a,b) such that f(c)=N
y
xb
y=N
a
(7) THEOREM If f(x) is continuous at b and , then
bg(x)ax
lim).lim()()(lim g(x)fbf)xf(g
axax
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Example Show that there is a root of the equation
4x3- 6x2 + 3x -2 =0 between 1 and 2.
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1.6 Tangent, and Other Rates of Change
A. Tangent
(1) Definition The Tangent line to the curve y=f(x) at point P( a, f(a)) is the line through P with slope
provided that this limit exists.ax
afxfm
ax
)()(lim
Example Find the equation of the tangent line to the parabola y=x2 at the point P(1,1).
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B. Other rates of change
The difference quotient
is called the average rate change of y with respect x over the interval [x1 , x2].
12
12 )()(xx
xfxfxy
(4) instantaneous rate of change=
at point P(x1, f(x1)) with respect to x.12
12
0
)()(limlim
12 xxxfxf
xy
xxx
Suppose y is a quantity that depends on another quantity x. Thus y is a function of x and we write y=f(x). If x changes from x1 and x2, then the change in x (also called the increment of x) is x= x2 - x1 and the corresponding change in y is x= f(x2) - f(x1) .
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(1) what is a tangent to a circle?
Can we copy the definition of the tangent to a circle by replacing circle by curve?
1.1 The tangent and velocity problems
The tangent to a circle is a line which intersects
the circle once and only once.
How to give the definition of tangent line to a curve?
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Fig. (a)
In Fig. (b) there are straight lines which touch the given curve, but they seem to be different from the tangent
to the circle.
L2
Fig. (b)L1
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Let us see the tangent to a circle as a moving line to a certain
line:
So we can think the tangent to a curve is the line approached by moving secant lines.
PQ
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Q'
x mPQ 2 3 1.5 2.5 1.1 2.1 1.01 2.01 1.001 2.001
Example 1: Find the equation of the tangent line to a parabola y=x2 at point (1,1). Q is a point on the curve.
Q
y=x2
P
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Then we can say that the slope m of the tangent line is the limit of the slopes mQP of the secants lines. And we express this symbolically by writing
mmQPPQ
lim
And
211
lim2
1
xx
x
So we can guess that slope of the tangent to the parabola at (1,1) is very closed to 2, actually it is 2. Then the equation of the tangent line to the parabola is
y-1=2(x-2) i.e y=2x-3.
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Suppose that a ball is dropped from the upper observation deck of the Oriental Pearl Tower in Shanghai, 280m above the ground. Find the velocity of the ball after 5 seconds.
elapsedtime
travelleddistancevelocityaverage
From physics we know that the distance fallen after t seconds is denoted by s(t) and measured in meters, so we have s(t)=4.9t2. How to find the velocity at t=5?
(2) The velocity problem:
Solution
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So we can approximate the desired quantity by computing the average velocity over the brief time interval of the n-th of a second from t=5, such as, the tenth, twenty-th and so on. Then we have the table:
Time interval Average velocity(m/s)
5<t<6 53.9
5<t<5.1 49.49
5<t<5.05 49.245
5<t<5.01 49.049
5<t<5.001 49.0049
The above table shows us the results of similar calculations of average velocity over successively smaller time periods.
It also appears that as time period tends to 0, the average velocity is becoming closer to 49. So the instantaneous velocity at t=5 is defined to be the limiting value of these average velocities over shorter time periods that start at t=5.
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1.2 The Limit of a Function Let us investigate the behavior of the function y=f(x)=x2-x+2 for values of x near 2.
x f(x) x f(x) 1.0 2.000000 3.0 8.0000001.5 2.750000 2.5 5.7500001.8 3.440000 2.2 4.6400001.9 3.710000 2.1 4.3100001.95 3.852500 2.05 4.1525001.99 3.970100 2.01 4.0301001.995 3.985025 2.001 4.003001
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We see that when x is close to 2(x>2 or x<2), f(x) is close to 4. Then we can say that: the limit of the function
f(x)=x2-x+2 as x approaches 2 is equal to 4.
Then we give a notation for this :
4)2(lim 2
2
xx
x
In general, the following notation:
(1) Definition: We write Lxfax
)(lim
Guess the value of .11
lim 21
xx
x
Notice that the function is not defined at x=1, and
x<1 f(x) x>1 f(x)
0.5 0.666667 1.5 0.4000000.9 0.526316 1.1 0.4761900.99 0.502513 1.01 0.497512 0.999 0.500250 1.001 0.4997500.999. 0.500025 1.0001 0.499975
Example 1
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and say “the limit of f(x), as x approaches a, equals L”.
Solution
If we can make the values of f(x) arbitrarily close to L (as close to
L as we like)by taking x to be sufficiently close to a but not equal to a.
Sometimes we use notation f(x) L as x a.
Example 1 Find4
59lim
2
4
tt
t
Example 2 Findx
xx
sinlim
0
Notice that as x a which means that x approaches a, x may >a and x may <a.
Example 3 Discuss , where )(lim0
xHx
01
00)(
xif
xifxH
The function H(x) approaches 0 as x approaches 0 and x<0,
and it approaches 1, as x approaches 0 and x>0.
So we can not say H(x) approaches a number as x a.
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One -side Limits:
Even though there is no single number that H(x) approaches as t approaches 0. that is, does not exist.)(lim
0xH
x
But as t approaches 0 from left, t<0, H(x) approaches 0. Then we can indicate this situation symbolically by writing:
0)(lim0
xHx
But as t approaches 0 from right, t>0, H(x) approaches 1. Then we can indicate this situation symbolically by writing:
)x(Hlimx
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We write
Lxfax
)(lim
And say the left-hand limit of f(x) as x approaches a (or the limit of f(x) as x approaches a from left) is equal to L. That is, we can make the value of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x less than a.
And say the right-hand limit of f(x) as x approaches a (or the limit of f(x) as x approaches a from right) is equal to L. That is, we can make the value of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x greater than a.
We write
Lxfax
)(lim
Here x a+ ” means that x approaches a and x>a.
(2)Definition:
Here x a- ” means that x approaches a and x<a.
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See following Figure:
What will it happen as x a or x b?
xOa b
y=f(x)
y
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(3)Theorem: if and only if
Lxfax
)(lim
20
1lim
xx
Lxfax
)(lim
Example: Find .
x 1/x2
±1 1±0.5 4±0.2 25±0.1 100±0,05 400±0,01 10000±0.001 1000000
Lxfax
)(lim
x
y=1/x2
O
y
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To indicate the kind of behavior exhibited in this example, we use the notation:
20
1lim
xx
Generally we can give following
Example Find ||lnlim0
xx
The another notation for this is f(x) as x a, which is read as “the limit of f(x), as x approaches a, is infinity” or “f(x) becomes infinity as x approaches a” or “f(x) increases without bound as x approaches a” .
f(x)limax
(4)DEFINITION: Let f be a function on both sides of a, except possibly at a itself. Then means that values of f(x) can be made arbitrarily large ( as we please) by taking x sufficiently close to a ( but not equal to a).
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y=f(x)=ln|x|
y
x
Obviously f(x)=ln|x| becomes large negative as x gets close to 0.
(5)DEFINITION: Let f be a function on both sides of a, except possibly at a itself.Then means that values of f(x) can be made arbitrarily large ( as we please) by taking x sufficiently close to a ( but not equal to a).
f(x)limax
The another notation for this is f(x) - as x a, which is read as “the limit of f(x), as x approaches a, is negative infinity” or “f(x) becomes negative infinite as x approaches a” or “f(x) decreases without bound as x approaches a” .
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(6)DEFINITION: The line x=a is called a vertical asymptote of the curve y=f(x) if at least one of the following statements is true:
Similar definitions can be given for one-side infinite limits.
)(lim xfax
)(lim xfax
)(lim xfax
)(lim xfax
Remember the meanings of x a- and x a+ .
)(lim xfax
)(lim xfax
)(lim xfax
)(lim xfax
)(lim xfax
)(lim xfax
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Find1
1lim
1 xx 11
lim1 xx
Example and
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1.3 Calculating limits using limit lawsLIMIT LAWS Suppose that c is a constant and the limits
exist. Thenand )(lim xgax
)(lim xfax
)(lim xfax
)]()([lim xgxfax
)(lim xgax
1.
)]()([lim xgxfax
)(lim xfax
)(lim xgax
2.
)](c[lim xgax
)(limc xgax
3.
g(x)
f(x)lim
ax )(lim
)(lim
xg
xf
ax
ax
0)(limif
xgax5.
)](f(x)[lim xgax
)(lim xfax
)(lim xgax4
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n
axxf ])([lim
(if n is even,we assume that )0f(x)lim ax
,lim ccax
,lim axax
nn
axax
lim
nn
axax
lim
nax
n
axxfxf )(lim)(lim
n
axxf ])(lim[
6. where n is a positive integer,
7.
8.
9 where n is a positive integer,
10. where n is a positive integer,
(if n is even, we assume that a>0)
11. where n is a positive integer,
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Example 6. Calculate
)1743(lim 23
3
xxx
x
1025911
lim 5
24
2
xxxx
x
])2(102[lim 33 3
1
xxx
x
12344
lim2
21
x
xx
x
11
lim22
1
xxxx
x
)(lim1
xgx
01
00
11
)(2 xx
x
xx
xg
Example 1. Find
Example 2. Find
Example 3. Calculate
Example 4. Calculate
Example 5. Calculate
where
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If f(x) is a polynomial or rational function and a is in the domain of
f(x), then .lim f(a)f(x)ax
(1) THEOREM if and only ifLxfax
)(lim Lxfax
)(lim )(lim xfax
Example : Show that .0||lim0
xx
Example: If
85
81)(
xx
xxxf ,determine whether
exists.
f(x)x 8lim
Example: Prove that xx
x
||lim
0does not exists.
Example: Prove that xx 2lim does not exists, where value of [x] is
defined as the largest integer that is less than or equal to x.
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(2) THEOREM If f(x) g(x) for all x in an open interval that contains a (except possibly at a) and the limits of f and g exist as x approaches a, then .limlim g(x)f(x)
axax
(3)SQUEEZE THEOREM If f(x) g(x) h(x) for all x in an open interval that contains a (except possibly at a) and then
,limlim Lh(x)f(x)axax
.lim Lg(x)ax
Example: Show that .0)1
arctan(lim0
x
xx
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