precise definition of limits

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