precise definition of limits
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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. - PowerPoint PPT PresentationTRANSCRIPT
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
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
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+).
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
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
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,
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}.
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
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
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.
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.
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
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
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
Negative infinity means
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
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
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
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.
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
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
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
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
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
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
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
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
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