LIMITSOF

FUNCTIONS

LIMITS OF FUNCTIONSOBJECTIVES:•define limits;•illustrate limits and its theorems; and•evaluate limits applying the given theorems.• define one-sided limits• illustrate one-sided limits• investigate the limit if it exist or not using

the concept of one-sided limits.•define limits at infinity;•illustrate the limits at infinity; and•determine the horizontal asymptote.

DEFINITION: LIMITS The most basic use of limits is to describe how a function behaves as the independent variable approaches a given value. For example let us examine the behavior of the function for x-values closer and closer to 2. It is evident from the graph and the table in the next slide that the values of f(x) get closer and closer to 3 as the values of x are selected closer and closer to 2 on either the left or right side of 2. We describe this by saying that the “limit of is 3 as x approaches 2 from either side,” we write

1xx)x(f 2

1xx)x(f 2

31xxlim 2

2x

2

3

f(x)

f(x)

x

y

1xxy 2

x 1.9 1.95 1.99 1.995 1.999 2 2.001 2.005 2.01 2.05 2.1

F(x) 2.71 2.852 2.97 2.985 2.997 3.003 3.015 3.031 3.152 3.31

left side right side

O

1.1.1 (p. 70) Limits (An Informal View)

This leads us to the following general idea.

EXAMPLEUse numerical evidence to make a conjecture about the value of .

1x

1xlim

1x

Although the function is undefined at x=1, this has no bearing on the limit. The table shows sample x-values approaching 1 from the left side and from the right side. In both cases the corresponding values of f(x) appear to get closer and closer to 2, and hence we conjecture that and is consistent with the graph of f.

1x

1x)x(f

21x

1xlim

1x

Figure 1.1.9 (p. 71)

x .99 .999 .9999 .99999 1 1.00001 1.0001 1.001 1.01

F(x) 1.9949 1.9995 1.99995 1.999995 2.000005 2.00005 2.0005 2.004915

THEOREMS ON LIMITS

Our strategy for finding limits algebraically has two parts:•First we will obtain the limits of some simpler function•Then we will develop a list of theorems that will enable us to use the limits of simple functions as building blocks for finding limits of more complicated functions.

We start with the following basic theorems, which are illustrated in Fig 1.2.1

Theorem 1.2.1 (p. 80)

axlim b kklim a

numbers. real be k and a Let Theorem 1.2.1

axax

Figure 1.2.1 (p. 80)

33lim 33lim 33lim

example, For

a. of values all for ax as kf(x)

why explains whichvaries, x as k at fixed remain

f(x) of values the then function, constant a is k xf If

x0x-25x

Example 1.

xlim 2xlim 0xlim

example, For

. axf that true be also must it ax then x, xf If

x-2x0x

Example 2.

Theorem 1.2.2 (p. 81)

The following theorem will be our basic tool for finding limits algebraically.

This theorem can be stated informally as follows:

a) The limit of a sum is the sum of the limits.b) The limit of a difference is the difference of the limits.c) The limits of a product is the product of the limits.d)The limits of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.e) The limit of the nth root is the nth root of the limit.

•A constant factor can be moved through a limit symbol.

5x2lim .14x

12x6lim .23x

)2x5(x4lim .33x

EXAMPLE : Evaluate the following limits.

31

58

5)4(2

5limxlim2

5limx2lim

4x4x

4x4x

6

12-18

12)3(6

12limx6lim3x3x

13

131

2)3(534

2limxlim5xlim4lim

2limx5limxlim4lim

2x5limx4lim

3x3x3x3x

3x3x3x3x

3x3x

4x5

x2lim .4

5x

3

3x6x3lim .5

3x

1x8lim .6

1x

21

10

425

52

4limxlim5

x lim2

4limx5lim

x2 lim

5x5x

5x

5x5x

5x

3375

15633

6limxlim3

6limx3lim

6x3lim

33

3

3x3x

3

3x3x

3

3x

2

3

4

9

3x

1x8lim

1x

OR

When evaluating the limit of a function at a given value, simply replace the variable by the indicated limit then solve for the value of the function:

22

3lim 3 4 1 3 3 4 3 1

27 12 1

38

xx x

EXAMPLE: Evaluate the following limits.

2x

8xlim .1

3

2x

Solution:

0

0

0

88

22

82

2x

8xlim

33

2x

Equivalent function:

(indeterminate)

2x

4x2x2xlim

2

2x

12444

4222

4x2xlim

2

2

2x

122x

8xlim

3

2x

Note: In evaluating a limit of a quotient which reduces to , simplify the fraction. Just remove the common factor in the numerator and denominator which makes the quotient . To do this use factoring or rationalizing the numerator or denominator, wherever the radical is.

0

0

0

0

x

22xlim .2

0x

Solution:

Rationalizing the numerator:

(indeterminate)0

0

0

220

x

22xlim

0x

22xx

22xlim

22x

22x

x

22xlim

0x0x

4

2

22

1

22

1

22x

1lim

22xx

xlim

0x0x

4

2

x

22xlim

0x

9x4

27x8lim .3

2

3

2

3x

Solution:

By Factoring:

(indeterminate)32

3

3

22

38 27

8 27 27 27 02lim

4 9 9 9 034 9

2

x

x

x

3

23

2

923

623

4

3x2

9x6x4lim

3x23x2

9x6x43x2lim

2

2

2

3x

2

2

3x

2

23

2

3

2

9

6

27

33

999

2

23

9x4

27x8lim

2

3

2

3x

5x

3x2xlim .4

2

3

2x

Solution:

33

222

2 2 2 32 3lim

5 2 5

8 4 3

4 5

15

9

15

3

x

x x

x

3

15

5x

3x2xlim

2

3

2x

DEFINITION: One-Sided Limits

The limit of a function is called two-sided limit if it requires the values of f(x) to get closer and closer to a number as the values of x are taken from either side of x=a. However some functions exhibit different behaviors on the two sides of an x-value a in which case it is necessary to distinguish whether the values of x near a are on the left side or on the right side of a for purposes of investigating limiting behavior.

Consider the function

0x ,1

0x ,1

x

x)x(f

1

-1

As x approaches 0 from the right, the values of f(x) approach a limit of 1, and similarly , as x approaches 0 from the left, the values of f(x) approach a limit of -1.

1xx

lim and 1xx

lim

,symbols In

oxox

1.1.2 (p. 72) One-Sided Limits (An Informal View)

This leads to the general idea of a one-sided limit

1.1.3 (p. 73) The Relationship Between One-Sided and Two-Sided Limits

EXAMPLE:

x

x)x(f 1. Find if the two sided limits exist given

1

-1

exist. not does xx

lim or

exist not does itlim sided two the thenxx

limxx

lim the cesin

1xx

lim and 1xx

lim

ox

oxox

oxox

SOLUTION

EXAMPLE:2. For the functions in Fig 1.1.13, find the one-sided limit and the two-sided limits at x=a if they exists.

The functions in all three figures have the same one-sided limits as , since the functions are Identical, except at x=a.

ax

1)x(flim and 3)x(flim

are itslim These

axax

In all three cases the two-sided limit does not exist as because the one sided limits are not equal. ax

SOLUTION

Figure 1.1.13 (p. 73)

3. Find if the two-sided limit exists and sketch the graph of

2

6+x if x < -2( ) =

x if x -2g x

4

26

x6lim)x(glim.a2x2x

4

2-

xlim)x(glim.b

2

2

2x2x

4)x(glim or4 to equal is and exist itlim sided two the then

)x(glim)x(glim the cesin

2x

2x2x

SOLUTION

EXAMPLE:

x-2-6 4

y

4

4. Find if the two-sided limit exists and sketch the graph of and sketch the graph.

2

2

3 + x if x < -2

( ) = 0 if x = -2

11 - x if x > -2

f x

SOLUTION

7

23

x3lim)x(flim.a

2

2

2x2x

7

2-11

x11lim)x(flim.b

2

2

2x2x

7)x(flim or

7 to equal is and exist itlim sided two the then

)x(flim)x(flimthe cesin

2x

2x2x

EXAMPLE:

graph. the sketch and

,exist f(x) lim if eminerdet ,4x23)x(f If .52x

3

4223

4x23lim )x(flim .a2x2x

3

4223

4x23lim )x(flim .b2x2x

3)x(flim or

3 to equal is and exist itlimsided two the then

)x(flim)x(flimthe cesin

2x

2x2x

SOLUTION

EXAMPLE:

f(x)

x

(2,3)

2

DEFINITION: LIMITS AT INFINITY

The behavior of a function as x increases or decreases without bound is sometimes called the end behavior of the function.

)x(f

If the values of the variable x increase without bound, then we write , and if the values of x decrease without bound, then we write .

xx

For example ,

0x

1lim and 0

x

1lim

xx

x

x

0x

1limx

0x

1limx

1.3.1 (p. 89) Limits at Infinity (An Informal View)

In general, we will use the following notation.

Figure 1.3.2 (p. 89)

Fig.1.3.2 illustrates the end behavior of the function f when L)x(flim or L)x(flim

xx

Figure 1.3.4 (p. 90)

EXAMPLEFig.1.3.2 illustrates the graph of . As suggested by this graph,

x

x

11y

ex

11lim

and ex

11lim

x

x

x

x

EXAMPLE ( Examples 7-11 from pages 92-95)

6x32x

lim .4

x311x2x5

lim .3

5x2xx4

lim .2

8x65x3

lim .1

2

x

23

x

3

2

x

x

336

x

36

x

xx5xlim .6

x5xlim .5

EXERCISES:

5w4w

7w7wlim 10.

2x

8xlim .5

19x9x2lim 9. 4y

y8y4lim .4

1y2y

3y2y1ylim 8.

1x

4x3xlim .3

1x

3x2x3x2lim 7.

4x3x

1x2lim .2

1x9

1x3lim 6. 2x5x4lim .1

2

2

1w

3

2x

2

134

5x

3

13

2y

2

2

1y3

2

1x

2

23

1x21x

2

3

1x

2

3x

A. Evaluate the following limits.

EXERCISES:

B. Sketch the graph of the following functions and the indicated limit if it exists. find

.

)x(glim.c g(x) lim.b g(x)lim.a

1x if 2x-7

1x if 2

1x if 3x2

)x(g .2

)x(flim.c f(x) lim.b f(x)lim.a

4- x if 4x

-4x if x4)x(f.1

1x1x1x

4x4x4x

.

)x(flim.c f(x) lim.b f(x)lim.a

1x2)x(g .5

)x(flim.c f(x) lim.b f(x)lim.a

x4)x(g .4

)x(flim.c f(x) lim.b f(x)lim.a

0x if 3

0x if x)x(f.3

2

1x

2

1x

2

1x

4x4x4x

0x0x0x