LIMITS 2. LIMITS 2.4 The Precise Definition of a Limit In this section, we will: Define a limit precisely.

LIMITSLIMITS

2

LIMITS

2.4The Precise Definition

of a LimitIn this section, we will:

Define a limit precisely.

Let f 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 every number , there is

a number such that

lim ( )x a

f x L

0

0

i 0 thenf ( )x a f x L

PRECISE DEFINITION OF LIMIT Definition 2

Since |x - a| is the distance from x to a and

|f(x) - L| is the distance from f(x) to L, and

since can be arbitrarily small, the definition

can be expressed in words as follows. the distance between f(x)

and L can be made arbitrarily small by taking the distance from x to a sufficiently small (but not 0).

Alternatively, the values of f(x) can be made as close as we please to L by taking x close enough to a (but not equal to a).

lim ( )x a

f x L

PRECISE DEFINITION OF LIMIT – in terms of distance

lim ( )x a

f x L

Therefore, in terms of intervals, Definition 2

can be stated as follows.

for every

(no matter how small is), we can find

such that, if x lies in the open interval

and , then f(x) lies in the open interval .

lim ( )x a

f x L

0 0

,a a

x a ,L L

PRECISE DEFINITION OF LIMIT – in terms of interval

We interpret this statement geometrically

by representing a function by an arrow

diagram as in the figure, where f maps

a subset of onto another subset of .

PRECISE DEFINITION OF LIMIT

Figure 2.4.2, p. 89

The definition of limit states that, if any small

interval is given around L, then

we can find an interval around a

such that f maps all the points in

(except possibly a) into the interval .

( , )L L

,a a


,a a

( , )L L

Figure 2.4.3, p. 89

If is given, then we draw

the horizontal lines and

and the graph of f.

0 y L y L

PRECISE DEFINITION OF LIMIT – in terms of graph

Figure 2.4.4, p. 89

If , then we can find a number

such that, if we restrict x to lie in the interval

and take , then the curve

y = f(x) lies between

the lines

and .

if such a has been

found, then any smaller

will also work.

lim ( )x a

f x L

0

,a a x a

y L y L


Figure 2.4.5, p. 89

The three figures show that, if a smaller is chosen, then a smaller may be required.


Figure 2.4.4, p. 89 Figure 2.4.5, p. 89 Figure 2.4.6, p. 89

Use a graph to find a number

such that

In other words, find a number that corresponds to in the definition of a limit for the function with a = 1 and L = 2.

3if 1 then 5 6 2 0.2x x x

0.2

3( ) 5 6f x x x

PRECISE DEFINITION OF LIMIT Example 1

Rewrite the inequality into graph the curves , y = 1.8, and y = 2.2 near

the point (1, 2).

estimate the x-coordinate of intersections are

about 0.911 and 1.124

Solution: Example 131.8 5 6 2.2x x

Figure 2.4.7, p. 89

3 5 6y x x

So, rounding to be safe, we can say that

This interval (0.92, 1.12) is not symmetric about x = 1.( left distance = 0.08, right distance = 0.12 )

Choose to be the

smaller distance, that is,

Assure the inequality

3if 0.92 1.12 then 1.8 5 6 2.2x x x

Solution: Example 1

Figure 2.4.8, p. 89

0.08

if 1 0.08x

3then 5 6 2 0.2x x

Prove that:

3lim(4 5) 7x

x


• Let be a given positive number. We want to find a number such that

• However,• Therefore, we want

• That is,

• This suggests that we should choose

if 0 3 then 4 5 7x x

4 5 7 4 12 4 3 4 3x x x x

Proof: Example 2

if 0 3 then 4 3x x

if 0 3 then 34

x x

4

showing that this works. Given , choose . If , then

Thus, Therefore, by the definition of a limit,

0

4

0 3x

4 5 7 4 12 4 3 4 44

x x x

if 0 3 then 4 5 7x x

PROOF Example 2

3

lim 4 5 7x

x

The example is illustrated by

the figure.

Figure Example 2

Figure 2.4.9, p. 91

Left-hand limit is defined as follows.


a number such that

Notice that Definition 3 is the same as Definition 2 except that x is restricted to lie in the left half of the interval .

lim ( )x a

f x L

0

0

if then ( )a x a f x L


,a a ,a a

Right-hand limit is defined as follows.


a number such that

In Definition 4, x is restricted to lie in the right half of the interval .

lim ( )x a

f x L

if then ( )a x a f x L


,a a ,a a

0

0

Use Definition 4

to prove that:

0lim 0x

x


Let be a given positive number.

Here, a = 0 and L = 0, so we want to find a number such that .

That is, . Squaring both sides of the inequality ,

we get . This suggests that we should choose .

if 0 then 0x x

Example 3STEP 1: GUESSING THE VALUE

if 0 thenx x x

2if 0 thenx x 2

Given , let .

If , then .

So, .

According to Definition 4, this shows that

0 2

0 x 2x

0x

0lim 0.x

x

STEP 2: PROOF Example 3

Prove that:2

3lim 9xx


Let be given. We have to find a number such that

To connect with we write

Then, we want

0 0 2if 0 3 then 9x x

STEP 1: GUESSING THE VALUE Example 4

2 9x 3x 2 9 ( 3)( 3)x x x

if 0 3 then 3 || 3x x x

Since

Thus we have

So , if x is chose 1 distance from 3

And

3 1 2 4x x

5 3 7x

3 7x


3 3 7 3x x x

However, now, there are two restrictions

on , namely

and

To make sure that both inequalities are satisfied, we take to be the smaller of the two numbers 1 and .

The notation for this is .

3x 3 1x 3

7x

C

7

min 1, 7


Given , let .

If , then (as in part l).

We also have , so

This shows that .

0 min 1, 7

0 3x 3 1 2 4 3 7x x x

3 7x 2 9 3 3 7

7x x x

2

3lim 9xx

STEP 2: PROOF Example 4

Using definition, we prove the

Sum Law. If and both exist, thenlim ( )

x af x L

lim ( )

x ag x M

lim ( ) ( )x a

f x g x L M


-

Let be given. We must find such that

0

0

if 0 then ( ) ( ) ( )x a f x g x L M

PROOF OF THE SUM LAW

Using the Triangle Inequality

we can write:

a b a b

( ) ( ) ( ) ( ( ) ) ( ( ) )

( ) ( )

f x g x L M f x L g x M

f x L g x M

PROOF OF THE SUM LAW Definition 5

We make less than

by making each of the terms

and less than .

Since and , there exists a number such that

Similarly, since , there exists a number such that

( ) ( ) ( )f x g x L M ( )f x L

( )g x M 2

02 lim ( )

x af x L

1 0 1if 0 then ( )

2x a f x L

lim ( )x ag x M

2 0 2if 0 then ( )

2x a g x M


Let .

Notice that

So, and

Therefore, by Definition 5,

1 2if 0 then 0 and 0x a x a x a

( )2

f x L

( )2

g x M

( ) ( ) ( ) ( ) ( )

2 2

f x g x L M f x L g x M


1 2min ,

To summarize,

Thus, by the definition of a limit,

if 0 then ( ) ( ) ( )x a f x g x L M

lim ( ) ( )x a

f x g x L M





Then, means that, for every

positive number M, there is a positive

number such that

lim ( )x a

f x

if 0 then ( )x a f x M

INFINITE LIMITS Definition 6

A geometric illustration is shown

in the figure. Given any horizontal line y = M, we can find a number

such that, if we restrict x to lie in the interval but , then the curve y = f(x) lies above the line y = M.

You can see that, if a larger M is chosen, then a smaller may be required.

0 ,a a x a

INFINITE LIMITS

Figure 2.4.10, p. 94

Use Definition 6 to prove that

Let M be a given positive number. We want to find a number such that

However,

So, if we choose and , then .

This shows that as .

20

1lim .x x

INFINITE LIMITS Example 5

2

1if 0 thenx Mx

22

1 1 1M x x

Mx M

1M

10 xM

21 Mx

2

1

x 0x




Then, means that, for every

negative number N, there is a positive

number such that

lim ( )x a

f x

if 0 then ( )x a f x N

INFINITE LIMITS Definition 7

This is illustrated by

the figure.

INFINITE LIMITS

Figure 2.4.11, p. 94

LIMITS 2. LIMITS 2.4 The Precise Definition of a Limit In this section, we will: Define a limit precisely.

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