Precise definition of limits
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![Page 1: Precise definition of limits](https://reader036.fdocuments.us/reader036/viewer/2022062308/56812f31550346895d94c41f/html5/thumbnails/1.jpg)
Precise definition of limits The phrases “x is close to a” and “f(x) gets closer
and closer to L” are vague. since f(x) can be arbitrarily close to 5 as long as x approaches 3 sufficiently. How close to 3 does x have to be so that f(x)
differs from 5 by less than 0.1? Solving the inequality |(2x-1)-5|<0.1, we get |x-3|<0.05, i.e., we find a number =0.05 such that whenever |x-3|< we have |f(x)-5|<0.1
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definition of a limit If we change the number 0.1 to other smaller
numbers, we can find other s. Changing 0.1 to
any positive real number , we have the following Definition: We say that the limit of f(x) as x
approaches a is L, and we write if
for any number >0 there is a number >0 such that
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Remark expresses “arbitrarily” and expresses
“sufficiently” Generally depends on To prove a limit, finding is the key point means that for every >0 (no matter
how small is) we can find >0 such that if x lies in
the open interval (a-,a+) and xa then f(x) lies in
the open interval (L-,L+).
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Example
Ex. Prove that
Sol. We solve the question in two steps.
1. Preliminary analysis of the problem (deriving a
value for ). Let be a given positive number, we
want to find a number such that
But |(4x-5)-7|=|4x-12|=4|x-3|, therefore we want
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Example (cont.)
This suggests that we should choose =/.
2. Proof (showing the above works). Given choose If 0<|x-3|<, then
|(4x-5)-7|=|4x-12|=4|x-3|<4Thus
Therefore, by definition we have
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Example
Ex. Prove that
Sol. 1. Deriving a value for . Let >0 be given, we
want to find a number such that
Since |(x2-x+2)-4|=|x-2||x+1|,if we can find a positive
constant C such that |x+1|<C, then |x-2||x+1|<C|x-2|
and we can make C|x-2|< by taking |x-2|</CAs we
are only interested in values of x that are close to 2,
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Example (cont.)
it is reasonable to assume |x-2|<1. Then 1<x<3, so
2<x+1<4, and |x+1|<4. Thus we can choose C=4 for
the constant. But note that we have two restrictions on
|x-2|, namely, |x-2|<1 and |x-2|</C=/4. To make sure
both of the two inequalities are satisfied, we take to
be the smaller of 1 and /4. The notation for this is
=min{1,/4}.
2. Showing above works. Given >0, let =min{1,/4}.
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Example (cont.)
If 0<|x-2|<, then |x-2|<1) 1<x<3) |x+1|<4. We also
have |x-2|</4, so |(x2-x+2)-4|=|x-2||x+1|</4¢4=This
shows that
can be found by solving the inequality, but no need to solve the inequality: is not unique, finding one is enough
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Example
Ex. Prove that
Sol. For any given >0, we want to find a number >0
such that
By rationalization of numerator,
If we first restrict x to |x-4|<1, then 3<x<5 and
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Example (cont.)
Now we have and we can make
by taking Therefore
If >0 is given, let
When 0<|x-4|<we have firstly
and then
This completes the proof.
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Proof of uniqueness of limits(uniqueness) If and then K=L.
Proof. Let >0 be given, there is a number 1>0 such that
|f(x)-K|< whenever 0<|x-a|<1. On the other hand, there is
a number 2>0 such that |f(x)-L|<whenever 0<|x-a|<2.
Now put =min{1,2} and x0=a+Then|f(x0)-K|<
and |f(x0)-L|<. Thus |K-L|=|(f(x0)-K)-(f(x0)-L)|·|f(x0)-K|+
|f(x0)-L|<2. Since is arbitrary, |K-L|<2 implies K=L.
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definition of one-sided limits
Definition: If for any number >0 there is a number
>0 such that
then
Definition: If for any number >0 there is a number
>0 such that
then
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Useful notations 9 means “there exist”, 8 means “for any”. definition using notations
such that
there holds
,
0, 0, : 0 | | ,x x a
| ( ) | .f x L
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M- definition of infinite limits
Definition. means that
8 M>0, 9 >0, such that
whenever
Remark. M represents “arbitrarily large”
( )f x M 0 | | .x a
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Negative infinity means
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Continuity Definition A function f is continuous at a number a if
Remark The continuity of f at a requires three things:
1. f(a) is defined
2. The limit exists
3. The limit equals f(a)
otherwise, we say f is discontinuous at a.
).()(lim afxfax
)(lim xfax
)(lim xfax
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Continuity of essential functionsTheorem The following types of functions are continuous
at every number in their domains:
polynomials algebraic functions power functions
trigonometric functions inverse trigonometric functions
exponential functions logarithmic functions
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Example Ex. Find the limits:(a) (b)
Sol. (a)
(b)
)1
2
1
1(lim
21
xxx.
1lim
2
1
x
nxxx n
x
.2
1
1
1lim
1
1lim
1
21lim)
1
2
1
1(lim
1212121
xx
x
x
x
xx xxxx
.2
)1(21
)]1()1(1[lim
1
)1()1()1(lim
1lim
21
1
2
1
2
1
nnn
xxx
x
xxx
x
nxxx
nn
x
n
x
n
x
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Continuous on an interval A function f is continuous on an interval if
it is continuous at every number in the interval. If f is defined only on one side of an
endpoint of the interval, we understand
continuous at the endpoint to mean continuous
from the right or continuous from the left.
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Continuity of composite functions
Theorem If f is continuous at b and
then
In other words, If g is continuous at a and f is continuous at
g(a), then the composite function f(g(x)) is
continuous at a.
lim ( ) ,x a
g x b
lim ( ( )) ( ).x a
f g x f b
lim ( ( )) (lim ( )).x a x a
f g x f g x
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Property of continuous functions
The Intermediate Value Theorem If f is
continuous on the closed interval [a,b] and let
N be any number between f(a) and f(b), where
Then there exists a number c in
(a,b) such that f(c)=N.
( ) ( ).f a f b
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Example The intermediate value theorem is often
used to locate roots of equations. Ex. Show that there is a root of the equation
between 1 and 2. Sol. f(1)=-1<0, f(2)=12>0, there exists a
number c such that f(c)=0.
3 24 6 3 2 0x x x
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Limits at infinity
Definition means for every >0 there
exists a number N>0 such that |f(x)-L|< whenever
x>N.
means 8>0, 9 N>0, such that
|f(x)-L|< whenever x<-N.
Lxfx
)(lim
Lxfx
)(lim
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Properties All the properties for the limits as x! a hold true
for the limits as x!1 and Theorem If r>0 is a rational number, then
If r>0 is a rational number such that is defined for
all x, then
.x
1lim 0.
rx x
rx
1lim 0.
rx x
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Examples
Ex. Find the limits
(a) (b)
Sol. (a)
(b)
145
23lim
2
2
xx
xxx
)1(lim 2 xxx
.5
3
/1/45
/2/13lim
145
23lim
2
2
2
2
xx
xx
xx
xxxx
.01/11
/1lim
1
1lim)1(lim
22
2
x
x
xxxx
xxx
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Horizontal asymptoteDefinition The line y=L is called a horizontal asymptote if
either or
For instance, x-axis (y=0) is a horizontal asymptote of the
hyperbola y=1/x, since
The other example, both and
are horizontal asymptotes of
Lxfx
)(lim Lxfx
)(lim
.01
lim xx
/ 2y / 2y arctan .y x
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Infinite limits at infinity
Definition means 8 M>0, 9 N>0, such
that f(x)>M whenever x>N.
means 8 M>0, 9 N>0, such that
f(x)<-M whenever x>N.
Similarly, we can define and
)(lim xfx
)(lim xfx
limx
lim .x
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Homework 3 Section 2.4: 28, 36, 37, 43
Section 2.5: 16, 20, 36, 38, 42
Section 2.6: 24, 32, 43, 53
Page 181: 1, 2, 3, 5, 7