Maths Ch4 Eng

8/8/2019 Maths Ch4 Eng

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CHAPTER IV Limit of a Function

2008 Marco Scaramastra

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CHAPTER IV

The concept of Limit of a Function

The notion of the limit of a function is one of the most important concepts in

mathematics. Generally speaking, limits are used to explain the behaviour of a function

when the independent variable approaches a value a.

In many problems we are interested in the behaviour of a function f(x) in the vicinity of

a point x = a. Two important issues arise.

What is the value offwhen x = a?

What is the value offwhen x is very close to, yet different from a?

Example. Let f be the function defined by f(x) = 3x + 1 and lets try to understand the

behaviour offat the point x = 2 and near the point x = 2.

The first step is easy; if x = 2 the value of the function is f(2) = 3(2) + 1 = 7

To investigate the behaviour of f near x = 2 we need to evaluate the function for

values that approach (yet differ from) x = 2. The following table shows some

possible values for x and f(x)

x ...1.99 1.999 1.9999 ...... 2.0001 2.001 2.01......

f(x) = 3x + 1 ...6.97 6.997 6.9997 ...... 7.0003 7.003 7.03......

It seems clear from the table that as x approaches 2 from either side, the value of f(x)

gets closer and closer to 7. The value 7 is called limitoff(x) as x approaches 2from

either side and we write

( )limx

x

+ =2

3 1 7

In this expression x 2 indicates the fact that x gets closer and closer to (yet differs

from) 2. Therefore, the definition of limit is the following.

Iff(x) becomes arbitrarily close to a single number L as x approaches a from either side,

then we write

( ) Lxflimax =

and say that the limit off(x) as x approaches a is L.


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For the function f(x) = 3x + 1 we saw that

( ) 7xlimand7)2(2x

==

That is, the value of the limit as x approaches 2 is the same as the value of the function

at x = 2. This is not always true! Lets consider the following example.

Example. Consider the following function and its limit

3x

9xlimand

3x

9x)x(

22

3x

=

First, observe that f(3) is undefinedsince substituting x = 3 in the formula for fleads

to a division by zero. However, as the following table suggests, there is a value for

the limit off(x) as x approaches 3, namely 6.

x ......2.99 2.999 2.9999 ...... 3.0001 3.001 3.01........

f(x) ......5.99 5.999 5.9999 ...... 6.0001 6.001 6.01........

To gain some geometric insight into this limit, lets change the function as follows. If

x 3, then we can write

3x3x

)3x)(3x(

3x

9x)x(

2

+=

+=

=

Thus, for x 3, the graph of f(x) coincides with the graph of the line y = x + 3;however for x = 3 there is no point on the graph since f(3) is undefined. The graph of

f is a straight line with a hole in it at x = 3, as in the figure, but it should be

intuitively evident that the limit off(x) as x 3 is equal to 6.

Example. Consider the following function and its limit

2x

1limand

2x

1)x(

2x =

First, observe that f(2) is undefinedsince substituting x = 2 in the formula for fleads

to a division by zero. However, as the following table shows, the values of f(x) get

O x3

y

6

yx

x=

29

3


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larger and larger as x 2 from the right side and get smaller and smaller as x 2

from the left side

x ......1,99 1,999 1,9999 ...... 2,0001 2,001 2,01........

f(x) ......-100 -1000 -10000 ...... 10000 1000 100........

Thus, f(x) does not approach any fixed number as x approaches 2, as the graph of the

function in the figure also shows.

If, as in the last example, the quantity f(x) approaches no single finite value as x

approaches a point a, then we say

( )xax

lim

does not exist.

O

x

yx

=

12

y

2


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4.1 Properties of Limits

Property 1

The limit of a constant function f(x) = k is the constant k; that is

limx a

k k

=

Property 2

limx a

x a

=

Property 3

The limit of a sum (of functions) is equal to the sum of the limits (of each function); the

limit of a difference (of functions) is equal to the difference of the limits (of each

function); the limit of a product (of functions) is equal to the product of the limits (of

each function). In formulas:

[ ]lim ( ) ( ) lim ( ) lim ( )x a x a x af x g x f x g x + = +

[ ]lim ( ) ( ) lim ( ) lim ( )x a x a x a

f x g x f x g x

=

[ ]lim ( ) ( ) lim ( ) lim ( )x a x a x a

f x g x f x g x

=

Property 4

A consequence of the rule for the limit of a product of functions is the following:[ ] =

n)x(flim

ax[ )x(flim

ax]n

nn )x(flim)x(flimaxax

=

Example.

[ ]lim lim limx x x

x x

+ = + = + =4 4 4

5 5 4 5 9

lim lim limx x xx x x = = =

2 2 2

2

2 2 4


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Example. If k is a constant, by using property 1 and 3 we can write

)x(flimk)x(kflim

or

)x(flimklim)x(kflim

axax

axaxax

=

=

The last example is an illustration of the fact that a constant multiple k can be moved out

of the operation with the limit.

Property5

The limit of a quotient (of functions) is equal to the quotient of the limits (of each

function), if the limit of the denominator is not zero; that is

0)x(glimif)x(glim

)x(flim

)x(g

)x(flim

axax

ax

ax=

Techniques for evaluating a limit

1) The limit of a polynomial function can be evaluated by direct substitution.

Formally we write

If p is a polynomial function and a is any real number, then

)a(p)x(plimax

=

2) In a previous example we saw that limx

x

x

=

3

2 9

36 . This result cannot be

obtained, though, with property 5 since we have limx

x

=3

3 0. However, by

factoring the quotient of functions we obtained the result. This leads to the

cancellation technique:

To calculate the limit of a quotient of functions when 0)(lim =

xgax

, we can

cancel any factors that are common to numerator and denominator, then calculate

the limit of the remaining expression (with property 5 if necessary).

3) A second way to calculate the limit of a quotient of functions when 0)(lim =

xgax

is called rationalising the numerator technique. For example, the following

limit:


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x

11xlim

0x

+

leads to the undefined form 0/0 if we use a direct substitution. In

this case we change the form of the fraction by rationalising the numerator:

11x1

)11x(xx

)11x(x1)1x(

11x11x

x11x

x11x

++=

++=

+++=

+++++=+

Therefore

x

11xlim

0x

+

=

11x

1lim

0x ++=

11

1

+=

2

1

One-sided limits

In the examples at the beginning of the chapter, we considered the empirical way to find

a limit. We substituted to the function both values approaching the value of x considered

in the limit from the right (just larger) and from the left (just smaller).

To denote these two possible approaches to a for the ( )xfax

lim , we use the symbols

( )xfax

lim and ( )xfax +

lim

In the first limit x approaches a from the left, while in the second one x approaches a

from the right. They are called respectively limit from the left and limit from the right.

The limits from the left and from the right can be different (as in the third example at the

beginning of the chapter). In that case, we said that the ( )xfax

lim does not exist.

Therefore we can state the following:

( ) Lxflimax

=

if and only if ( ) Lxf

ax

=

lim and ( ) Lxf

ax

=+

lim .

This result is always true for polynomial functions.

Infinite limits

It is possible that when we consider the limit from the left, f(x) increases without bound

and when we consider the limit from the right f(x) decreases without bound (or

viceversa). In other words, the ( )xfax

lim is an infinitely large positive or negative value,

that we denote with and . Obviously, in this case we also said that the ( )xfax

lim

does not exist and this type of limit is called an infinite limit.


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Iff(x) approaches infinity (or negative infinity) as x approaches a from the right or left,

then the vertical line x = a is called a vertical asymptote off(x).

Limits at infinity

It is common to check the behaviour of a function when x increases or decreases without

bound. In many cases, the function also increases or decreases without bound and the

value of the limit ( or ) can be checked by substituting to x a very large positive or

negative value. If the limit is instead a finite number, we have the following definition.

Iff(x) approaches a finite number L as x approaches infinity (positive or negative), the

line y =L is called a horizontal asymptote off(x).

The calculation of such a limit, however, can be very difficult.

Evaluating limits at infinity

The following theorem is useful:

Ifris a positive real number, then 01

lim = rxx

.

Furthermore, if xr

is defined when x < 0, then 01

lim = rxx

In general, if a rational function has a horizontal asymptote, then it must approach the

same asymptote to the right and to the left.

This is not true of other types of functions. For instance, the function1

)(2 +

=

x

xxf

has y = 1 as a horizontal asymptote to the right and y = 1 as a horizontal asymptote to

the left. The calculation of the limit is not easy. However, we can notice that as x

increases without bound, numerator and denominator tend to become the same number

(for large x the term +1 tend to lose its importance in the radical) and the ratio tends to

become +1 for positive values of x, and 1 for negative values of x.

Rational functions

There is an easy way to determine whether the graph of a rational function has a

horizontal asymptote. This shortcut is based on the comparison of the degrees of the


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numerator and the denominator of the rational function. The following examples

illustrate a method for evaluating the limit at infinity of similar rational functions.

(a)13

32lim

2 +

+

x

x

x(b)

13

32lim

2

2

+

+

x

x

x(c)

13

32lim

2

3

+

+

x

x

x

In this method we divide both numerator and denominator by the highest power of x.

(a) ( by x2)

13

32lim

2 +

+

x

xx = 2

2

/13

/3/2lim

x

xxx +

+

= 03

00

+

+=

3

0= 0

(b) ( by x2)

13

32lim

2

2

+

+

x

xx = 2

2

/13

/32lim

x

xx +

+

= 03

02

+

+=

3

2

(c) (by x3)

13

32lim

2

3

+

+

x

xx = 3

3

/1/3

/32lim

xx

xx +

+

= 00

02

+

+=

0

2= undefined

Observe that in part (a) the degree of the numerator was less than the degree of the

denominator and the limit of the ratio was 0. In part (b), the degrees of the numerator

and denominator were equal and the limit of the ratio was given by the ratio of the

coefficients of the highest-powered terms. Finally, in part (c), the degree of the

numerator was greater than that of the denominator and the limit of the ratio did not

exist (undefined). These results suggest the following definition.

Limits at infinity for rational functions

For the rational function given by y =f(x)/g(x), where

f(x) = anxn

+ an-1xn-1 + + a0 and g(x) = bmx

m+ bm-1x

m-1 + + b0 we have

>

=

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Maths Ch4 Eng