1

Function

Normally we may present a function via four common ways: 1) Numeric (tables) 2) Visual (graphs) 3) Description (words) 4) Algebra (formulas)

The method of representation often depends on how the function arises. Table 1 For example 1: Year t Speed S(mi/h)

1994 228.011 1995 231.604 1996 233.100 1997 218.263 1998 223.503 1999 225.179 2000 223.471 2001 226.037 2002 231.342 2003 231.725 2004 222.024 2005 227.598 2006 228.985 2007 225.817 2008 226.366 2009 224.864 2010 227.970 2011 227.472

Table 1 shows the top qualifying speed S for the Indianapolis 500 auto race as a indianapolis 500 qualifying speeds function of the year t. There is exactly one value of S for each value of t.

2

Example 2

Figure 1

Figure 1 is a graphical record of an earthquake recorded on a seismograph. The graph describes the deflection D of the seismograph needle as a function of the time T elapsed since the wave left the earthquake’s epicenter. There is exactly one value of D for each value of T . Example 3 Description (words) Some of the most familiar functions arise from formulas; for example, the formula

2C r expresses the circumference C of a circle as a function of its radius r . There is exactly one value of C for each value of r .

3

Example 4 Algebra (formulas) „ Sometimes functions are described in words. For example, Isaac Newton’s Law of Universal Gravitation is often stated as follows: “The gravitational force of attraction between two bodies in the Universe is directly proportional to the product of their masses and inversely proportional to the square of the distance between them”.

1 2

2

m mF G

r

This is the verbal description of the formula in which F is the force of attraction, m1 and m2 are the masses, r is the distance between them, and G is a constant.

If the masses are constant, then the verbal description defines F as a function of r. There is exactly one value of F for each value of r.

4

Definition 1.1 A function f is a rule that associates a unique output with each input.

If the input is denoted by x, then the output is denoted by y or f (x) (read “f of x”) see Figure 2.

Figure 2

In this definition the term unique means “exactly one.” Thus, a function cannot assign two different outputs to the same input.

Example 1 Table 2 describes a functional relationship y = f (x) for which Table 2

Example 2 The equation

23 4 2y x x has the form ( )y f x in which the function f is given by the formula

5

2( ) 3 4 2f x x x Graphs of functions If f is a real-valued function of a real variable, then the graph of f in the xy-plane is defined to be the graph of the equation ( )y f x . For example,

Figure 3

It is seen that, from Figure 3, graphs can provide valuable visual information about a function.

6

For example, since the graph of a function f in the xy-plane is the graph of the equation ( )y f x , the points on the graph of f are of the form

, x f x ; that is, the y-coordinate of a point on the graph of f is the value of f at the corresponding x-coordinate (Figure 4).

Figure 4 The y-coordinate of a point on the graph of y = f(x) is the value of f at the corresponding x-coordinate

The values of x for which ( ) 0f x are the x-coordinates of the points where the graph of f intersects the x-axis (Figure 5). These values are called the zeros of f , the roots of ( ) 0f x , or the x-intercepts of the graph of ( )y f x .

Figure 5 f has zeros at x1, 0, x2, and x3.

7

Vertical line test A curve in the xy-plane is the graph of some function f if and only if no vertical line intersects the curve more than once (see Figure 6).

Figure 6 This curve cannot bethe graph of a function

Example 3 The graph of the equation

2 2 25x y is a circle of radius 5 centered at the origin and hence there are vertical lines that cut the graph more than once (Figure 7). Thus this equation does not define y as a function of x

8

Figure 7 Absolute Value Function

Recall that the absolute value or magnitude of a real number x is defined by , 0

| | , 0

x xx

x x

And its graph is shown in Figure 8. The effect of taking the absolute value of a number is to strip away the minus sign if the

number is negative and to leave the number unchanged if it is nonnegative. Thus

4 45 5,  , 0 0

7 7

A function is a rule that takes certain numbers as inputs and assigns to each a definite output number. The set of all input numbers is called the domain of the function and the set of resulting output numbers is called the range of the function.

Piecewise-Defined functions The absolute value function f x x is an example of a function that is defined piecewise in the sense that the formula for f changes, depending on the value of x.

Figure 8

9

Example 4 Sketch the graph of the function defined piecewise by the formula Solution. The formula for f changes at the points 1x and 1x . (We call these the breakpoints for the formula.)

Figure 9

Domain and Range If x and y are related by the equation y = f (x), then the set of all allowable inputs (x-values) is called the domain of f , and the set of outputs (y-values) that result when x varies over the domain is called the range of f . Notations: Let f be a function from A to B ( :f A B ) fD represents domain of function f

fR represents range of function f

Image of x is y since ( )f x y :f A B is called a function from A onto B if fR B

10

Example 5 Consider a set ( 3,1),(0,2),(3, 1),(5,4) . Is it a function?

Domain: Range: Natural Domain Definition 1.2 If a real-valued function of a real variable is defined by a formula, and if no domain is stated explicitly, then it is to be understood that the domain consists of all real numbers for which the formula yields a real value. This is called the natural domain of the function. The domain and range of a function f can be pictured by projecting the graph of y = f(x) onto the coordinate axes as shown in Figure 9.

Figure 9 The projection of y = f(x) on the x-axis is the set of allowable x-values for f , and the projection on the y-axis is the set of corresponding y-values.

11

Example 6 1) siny x has all real number as a domain and range is [ 1,1] .

2) 2 4y x has all real number as a domain and range is [2, ) Example 7

Domain of 4( )

1f x

x

is

Domain of 2

( )9

xf x

x

is

Domain of 4( )

xf x

x

is

Domain of 2( ) 4f x x is Example 8

22 9 4, 4

( ) 4

5 , 4

x xx

h x x

x

or 2 1 , 4( )

5 , 4

x xh x

x

fD

fR

12

Equality Functions Definition 1.3 Function f equals to function g if and only if 1. f gD D 2. For all fx D , we have ( ) ( )f x g x

Example 9

1) Let 2 2( )

xf x

x

and 1

( )2 2

g xx

2) Let ( ) 3f x x and

22 7 3 1,

2 1 2( )5 1

,2 2

x xx

xg x

x

13

DOMAIN AND RANGE IN APPLIED PROBLEMS Example 10 An open box is to be made from a 16-inch by 30-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides (see Figure 10). (a) Let V be the volume of the box that results when the squares have sides of length x. Find a formula for V as a function of x. (b) Find the domain of V . (c) Use the graph of V given in Figure 10(c) to estimate the range of V . (d) Describe in words what the graph tells you about the volume.

Figure 10

14

Exercise

2.

15

3. 4.

5.

16

6. Find the natural domain and determine the range of each function.

17

ARITHMETIC OPERATIONS ON FUNCTIONS Definition 1.4 Given functions f and g, we define

For the functions f + g, f g , and fg we define the domain to be the intersection of the domains of f and g, and for the function f /g we define the domain to be the intersection of the domains of f and g but with the points where g(x) = 0 excluded (to avoid division by zero). Example 11 Let

Composite Function Definition 1.5 Let f and g be functions and g fR D o . Composite Function of f and g denoted by fog is a function ( )( ) ( ( ))fog x f g x whose domain is { : gx x D and ( ) }fg x D .

18

Example 12 Let ( ) 3f x x and ( ) 2 1g x x a) Let F fog Find ( )F x and domain of F b) Let G gof Find ( )G x and domain of G c) Let H fof Find ( )H x and domain of H Solutions a) Domain of g is ( , ) and domain of f is [3, )

( ) ( )( ) ( ( )) (2 1)F x fog x f g x f x 2 4x ( )g x is in domain of f if 2 1 3x and

Domain of F is a set of x where ( )g x is in domain of f Thus domain of F is a set of x where 2x i.e. [2, ) .

19

Symmetry Definition 1.6 Let f be a function 1. If ( ) ( )f x f x , f is called an odd function whose graph is

symmetric about the origin. 2. If ( ) ( )f x f x , f is called an even function whose graph is

symmetric about the y-axis.

20

Graph of Even and odd functions

LET the graph of a function is on the right hand side.

We will now draw the left-hand side -- so that the graph will be symmetrical with respect to the y-axis:

In this case,

f(−x) = f(x).

The height of the curve at −x is equal to the height of the curve at x -- for every x in the domain of f.

Again, let this be the right-hand side:

We will now draw the left-hand side -- so that the graph will be symmetrical with respect to the origin:

Every point on the right-hand side is reflected through the origin. In this case,

f(−x) = −f(x).

The value of f at −x is the negative of the value at x.

21

Example 13 a) Let 3( )f x x Consider 3( ) ( )f x x f x Thus f is an odd function shown in figure 1 below.

-4 -3 -2 -1 1 2 3 4

-5

5

x

y

Figure 11

b) Let 2( ) 3 1f x x Consider 2( ) 3 1 ( )f x x f x Thus f is an even function whose graph shown in Figure 2

-2 -1 1 2

-2

2

4

x

y

Figure 12

22

Example 14

This is the curve f(x) = x3−x+1

It is not an odd function, and it is not an even function either.

It is neither odd nor even!

Inverse function

An inverse function goes the other way! Consider the function f(x) = 2x+3, written as a flow diagram:

The Inverse Function just goes the other way:

So the inverse of: 2x+3 is: (y-3)/2

23

The inverse is usually shown by putting a little "-1" after the function name, like

this:

f-1(y)

We say "f inverse of y"

So, the inverse of f(x) = 2x+3 is written:

f-1(y) = (y-3)/2

(I also used y instead of x to show that we are using a different value.)

Note that the inverse is that it should give us back the original value:

When the function f turns the apple into a banana,

Then the inverse function f-1 turns the banana back to the apple. Example:

Using the formulas from above, we can start with x=4:

f(4) = 2×4+3 = 11

We can then use the inverse on the 11:

f-1(11) = (11-3)/2 = 4

And we magically get 4 back again!

We can write that in one line:

f-1( f(4) ) = 4

"f inverse of f of 4 equals 4"

24

No Inverse?

To be able to have an inverse we need unique values.

Let us see graphically what is going on here:

No Inverse

Inverse is Possible

When a y-value has more

than one x-value, how do we

know which x-value to go

back to?

When there is a unique y-

value for every x-value we

can always "go back" from y

to x.

So we have this idea of "a unique y-value for every x-value", and it actually has a

name. It is called "Injective" or "One-to-one":

When a function is "One-to-one", it has an inverse. One to one function Definition 2.1 Let f be a function. f is called a one to one function if and only if for all , ,x y z if ( , )x y and ( , )z y f then x z .

25

Inverse function Definition 2.2 Let f be a one to one function from A onto B . An inverse function of f is defined by 1 ( , ) ( , )f b a a b f which is also a one to one function from B to A.

Note also:

When we restrict the domain so there is

a unique y for every x ...

... and now we can have an inverse:

The function f(x) goes from the

domain to the range,

The inverse function f-1(y) goes from

the range back to the domain.

So, the f(x) and f-1(y) are like mirror images. In other words:

The graph of f(x) and f-1(x) are symmetric across the line y=x as shown

in Figure below.

26

f(x) and f-1(x) are like mirror images

(flipped about the diagonal).

You can see on this last picture that there is a definite graphical relationship between the points of the function and the points of the inverse. You can use this relationship if you're given a random graph and are told to graph the inverse.

Suppose you are given this graph:

Now draw the reflection line:

27

Now eyeball the graph, and draw the diagonals from known points on the graph to their "reflections" on the other side of the line:

Note that the points actually ON the line y = x don't move; that is, where the function crosses the diagonal, the inverse will cross, too. Now draw in some plot-points:

...and connect the dots:

28

Example Find an inverse of f where 3( ) 1f x x Solution From 3( ) 1y f x x i.e. 3 1x y So 1 3( , ) 1f y x y x or 1 3( , ) 1f x y y x

We normally write 1( )f x 3 1x so that we can easily draw graphs of both functions f and 1f as follows

-4 -3 -2 -1 1 2 3 4

-5

5

x

y

Figure 4

y x

1 3( ) 1f x x

3( ) 1f x x

29

Other Interesting Functions All functions here will be useful in the next sections. Algebraic Function a. Polynomial Function:

11 1 0( ) ...n n

n nf x a x a x a x a

where ia is a real number for each 0,1,2,...,i n n is a non-negative integer. If n is the largest number such that 0na , we call f a degree n polynomial function of such as 3 2( ) 3 5 4f x x x x is a degree 3 polynomial function.

30

b. Rational Function is a function formed by a ratio between two polynomial functions.

11 0

11 0

...( )

...

n nn n

m mm m

a x a x af x

b x b x b

,

where 11 0... 0m m

m mb x b x b

Example 11 Let

2

( )x x

y f xx

Rewrite f : ( ) 1f x x where 0x Thus graph of ( )f x is the graph of 1y x , but undefined at 0x

f(x)=(X^2+x)/x

-2 -1.5 -1 -0.5 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0.5

1

1.5

2

x

y

O

Figure 5

c. Functions of the form ( )n f x where ( )f x is either polynomial or rational function.

31

d. Functions formed by summation, multiplication and division of functions in part a. to c. Below are some examples of functions in part c. and d.

1) 23( )f x x 2) 4( )

1

xf x

x

3) ( )1

xf x

x

Example

32

Transcendental Functions a. Exponential Function For ,x

oy y a 1a , its graph is shown in Figure 6 ( 1oy ) For ,x

oy y a 0 1a , its graph is shown in Figure 7 ( 1oy )

-4 -3 -2 -1 1 2 3 4

-4

-2

2

4

x

y

Figure 6

-4 -3 -2 -1 1 2 3 4

-4

-2

2

4

x

y

Figure 7

b. Logarithmic Function Logarithmic function is an inverse of exponential function. Given an exponential function xy a ( 1oy ). Then its inverse function is

yx a or we can write it as logay x . If logay x , 1a , then its graph is shown in Figure 8. If logay x ,0 1a , then its graph is shown in Figure 9.

33

-4 -3 -2 -1 1 2 3 4

-4

-2

2

4

x

y

Figure 8

-4 -3 -2 -1 1 2 3 4

-4

-2

2

4

x

y

Figure 9

Properties of logarithmic functions 1. Domain of a logarithmic function is { : 0}x x and its range is { : }y y 2. A logarithmic function is a one-to-one function. 3. log 1 0a 4. Graph of logay x is a reflection of the graph xy a across the line y x .

34

Remark: When a e (where 2.71818...e = natural number) xy e has the inverse logey x which is normally written as lny x

and it is called a natural logarithm. The properties of xy e and lny x are the same as of the following properties of xy a and logay x ( 0a ), respectively

Properties of logarithmic and exponential functions Given positive numbers ,a b where 1, 1a b and ,x y R 1. x y x ya a a

2. x

x yy

aa

a

3. ( )x x xa b ab and xx

x

a a

bb

4. y

x xya a

5. 1xx

aa

6. If 0, 0x y , then log ( ) log loga a axy x y

log ( ) log loga a ax

x yy

7. log logra ax r x 8. log

loglog

ba

b

xx

a

9. log 1a a 10. ln xe x and ln xe x , 0x 11. xa y and logax y , 0y

35

Example 12 Find the values of x (a) 4 3 8 6x x (b) 2 177x xe

36

c. Trigonometric Function

siny x cosy x sintan

cos

xy x

x

1csc

siny x

x 1

seccos

y xx

coscot

xy x

sinx

radians

0

6

4

3

2

degree

0 30 45 60 90

sin 1

2

1

2 2 1

2 2

3

2

1

cos 3

2

3

2

1

2

1

2

0

37

38

39

Example Find the sine and cosine of 2 / 3 radians.

Example Find the sine and cosine of / 4 radians.

40

Example Complete the following table of the function

values.

41

42

Example Find the domain and range of the

following functions.

43

44

Graph of trigonometric functions

45

siny x cosy x sintan

cos

xy x

x

1csc

siny x

x 1

seccos

y xx

coscot

xy x

sinx

-2π -π π 2π 3π

-1

1

x

y

Graph of siny x

-2π -π π 2π 3π

-1

1

x

y

Graph of cosy x

-2π -π π 2π 3π

-5

5

x

y

Graph of tany x

46

-2π -π π 2π 3π

-5

5

x

y

Graph of coty x

-2π -π π 2π 3π

-5

5

x

y

Graph of secy x

-2π -π π 2π 3π

-5

5

x

y

Graph of cscy x Normally, the inverse of a trigonometric function is not a function

since each trigonometric function is not one-to-one. However, if we restrict the domain, we can make a one-to-one trigonometric function and define an inverse function as follows.

47

1) Restrict the domain of siny x to ,2 2

Its inverse function is arcsin y x .

-π -π/2 π/2 π

-1

1

x

y

-4 -3 -2 -1 1 2 3 4

-π/2

π/2

x

y

siny x arcsin y x 2) Restrict domain of cosy x to 0, Its inverse function is arccosy x .

-π -π/2 π/2 π

-1

1

x

y

-1.5 -1 -0.5 0.5 1 1.5

-π/2

π/2

π

x

y

cosy x arccosy x

48

3) Restrict domain of tany x to ,2 2

Its inverse function is arctany x .

-2π -3π/2 -π -π/2 π/2 π 3π/2 2π

-5

5

x

y

-6 -4 -2 2 4 6

-π

-π/2

π/2

π

x

y

tany x arctany x 4) Restrict domain of coty x to 0, Its inverse function is arccot y x .

-2π -3π/2 -π -π/2 π/2 π 3π/2 2π

-2

2

x

y

-4 -3 -2 -1 1 2 3 4

-π

-π/2

π/2

π

x

y

coty x arccot y x

49

5) Restrict domain of secy x to [0, ) ( , ]2 2

Its inverse function is arcsec y x .

-π -π/2 π/2 π

-4

-2

2

4

x

y

-4 -3 -2 -1 1 2 3 4

-π/2

π/2

π

x

y

secy x arcsec y x

6) Restrict domain of cscy x to [ ,0) (0, ]2 2

Its inverse function is arccsc y x .

-π -π/2 π/2 π

-4

-2

2

4

x

y

-4 -3 -2 -1 1 2 3 4

-π

-π/2

π/2

π

x

y

cscy x arccsc y x

50

Exercises on Functions 1. Determine if the following are functions. Locate domain and range.

(a) (1,3),(2,3),(3,4),(4,5) (b) ( , ) : 4 1x y y x (c) 4 1y x (d) Let

x y 15 2 2 13 13 13 5 3

2. Determine if each following function is either even or odd or neither. (a) 3( ) 2f x x x

(b) 2

8( )

2g x

x

(c) ( ) 3h x x x (d) ( )k x x x 3. What is the difference of 2 2sin ,sinx x and sin(sin )x ? Show in

terms of composite functions.

Answers to Function Exercises

51

1. (a) yes {1,2,3,4}D and {3,4,5}R (b) no D R all real numbers (c) yes D and { : 1}R y y (d) yes {2,5,13,15}D and {2,3,13}R 2. (a) odd (b) even (c) odd (d) neither 3. Let ( ) sinf x x and 2( )g x x

2sin ( ( )),x f g x 2sin ( ( )),x g f x while

sin(sin ) ( ( ))x f f x

52

Limit and Continuity of Function

2.1 Limit of function

Limit of a function ( )f x at x a is not the value of f a but it is a value of ( )f x as x approaches a . There are two types of a : real number and infinity. 2.1.1 Limit of function as x a (real number) Example Consider how a function behaves as the independent variable approaches a given value.

Suppose ( ) 5 1f x x , for some values of x approaching 1a shown in Table 1.

Table 1

x 0.5 0.9 0.99 0.999 1 1.001 1.01 1.1

( )f x 1.5 3.5 3.95 3.995 ... 4.005 4.05 4.5

( )g x 0 0 0 0 ... 1 1 1 Left side Right side

53

We can see that when x approaches 1a , ( )f x gets closer and closer to 4L . However, ( ) 1g x when 1x and ( ) 0g x when

1x . Thus ( )g x does not approach just one number. Therefore, we say ( )f x has limit equal to 4 as x approaching 1.

( )g x does not have a limit as x approaching 1. We write them as

1lim ( ) 4x

f x

and 1

lim ( )x

g x

does not exist.

-4 -3 -2 -1 1 2 3 4

-4

-2

2

4

x

y

( ) 5 1f x x

Graph of ( )f x shows that the value of ( )f x gets closer to 4 as x approaching 1.

We can define limit as the following:

Definition If ( )f x gets closer to L as x approaching a , we say that L is a limit of ( )f x when x approaches a denoted by lim ( )

x af x L

The values of x approaches a from two sides:

54

x approaches a from the right denoted by x a x approaches a from the left denoted by x a

From the example, we have

1lim 5 1

xx

1lim 5 1 4

xx

If a function has the same limit from both sides, we have (Right limit) lim ( )

x af x

(Left limit) lim ( )

x af x

lim ( )x a

f x

So we have the following theorem:

Theorem 1 lim ( )x a

f x

exists and equals to L if

(1) both lim ( )x a

f x

and lim ( )x a

f x

exist

(2) lim ( )x a

f x

lim ( )x a

f x

Example 1 Compare 0

limx

x

x and

2

0limx

x

x

Solution

55

Properties of limits Let , , ,a k L M be real numbers. lim ( )

x af x L

and lim ( )

x ag x M

1. lim ( )x a

kf x kL

2. lim [ ( ) ( )]x a

f x g x L M

3. lim ( ) ( )x a

f x g x LM

4. ( )lim , 0

( )x a

f x LM

g x M

5. If ( )f x is a polynomial function, then for any number a lim ( ) ( )

x af x f a

6. lim ( ) lim ( )nn

x a x ag x g x

7. lim sin sinx a

x a

lim csc cscx a

x a

lim cos cosx a

x a

lim sec secx a

x a

lim tan tanx a

x a

lim cot cotx a

x a

Example 2 3 2

0

3 4lim

cosx

x x

x

56

Example 3 Define

22 , 0

( ) ,0 1

1 , 1

x x

f x x x

x x

Find the limits of ( )f x as x approaching 0 and 1 Solution

Example 4 Evaluate 1

3 32

9lim 2 9 sin 2x

x x x

Solution

57

To find a limit, some time we replace x by a and get a function in

a form of 0

0. So we may use these two techniques.

1) Factoring 2) Conjugating

Example 5 Calculate 3 2

23

9 9lim

6x

x x x

x x

Solution

Example 6 Calculate 0

2lim

16 2 4x

x

x

Solution

58

If the two techniques do not help in solving 0

0, we may apply the

following theorem.

Squeeze Theorem If ( ) ( ) ( )f x g x h x for all values of ,x x a at some points a and lim ( )x a

f x

lim ( )x a

h x L

, then lim ( )x a

g x L

L

ax

y

g

f

h

Example 6 Evaluate the following limit.

2

0

1lim cos

x

xx

Example 7 Use Squeeze Theorem to show

52

2

0 4

lim 0

1 1x

x

x

59

Example 8

1. If 33 ( ) 2x f x x for 0 2x , evaluate 1

lim ( )x

f x

2. Calculate 2

0

2lim sinx

xx

Solution

Theorem

1. 0

sinlim 1x

x

x 2.

0

cos 1lim 0x

x

x

60

Example 9 Use 0

sinlim 1x

x

x to show

0

cos 1lim 0x

x

x

Proof 0 0

cos 1 cos 1 cos 1lim lim

cos 1x x

x x x

x x x

2

0

cos 1lim

(cos 1)x

x

x x

2

0

sinlim

(cos 1)x

x

x x

0

sin sinlim

cos 1x

x x

x x

0 0

sin sinlim lim 1 0 0

cos 1x x

x x

x x

Example 10 Evaluate 21

sin( 1)lim

2x

x

x x

Solution

61

2.1.2 Limit of function as x (limit at infinity) When domain of ( )y f x is unbounded. The values of ( )f x may get closer to one value as x increases unboundedly (written as x ) or x decreases unboundedly (written as x )

Consider function 1( )f x

x . Its graph is shown here.

-4 -3 -2 -1 1 2 3 4

-4

-2

2

4

x

y

Table 2

x 100 1000 10000 Increases

unboundedly 1

( )f xx

0.01 0.001 0.0001 ... 0

x -100 -1000 -10000 Decreases

unboundedly 1

( )f xx

-0.01 -0.001 -0.0001 ... 0

62

The values of ( )f x as x increasing or decreasing are shown in Table 2. When x , the values of ( )f x get closer to 0 and ( ) 0f x . So

we say that limit of ( )f x equals 0 as x denoted by 1lim 0

x x .

When x , the values of ( )f x get closer to 0 as well, but ( ) 0f x .

We say that limit of ( )f x equals 0 as x denoted by 1lim 0

x x

The graph above shows that 1( )f x

x gets closer to x -axis as x

increasing to infinity and decreasing to negative infinity, but never hit x -axis. We call this line the graph gets closer to an asymptote. Properties of limits at infinity Many properties are the same as those of limits at a finite number a . Let , ,k L M be real numbers. lim ( )

xf x L

and lim ( )

xg x M

1. limx

k k

2. lim [ ( ) ( )]x

f x g x L M

3. lim ( ) ( )x

f x g x LM

4. ( )lim , 0

( )x

f x LM

g x M

5. 1 1

lim [ ( )]n n

xf x L

where n is positive and 0L

6. 1lim 0

nx x where n is a positive integer

63

All 6 properties are the same when replacing x by x Example 1 Calculate

a) 3

5lim

x x b)

23

3lim

x x

Solution

64

Example 2 Evaluate 4 2

2

3 7 6lim

4 3 6x

x x

x x

Solution

Example 3 Evaluate 2 3

lim3x

x

x

Solution

65

Example 4 Calculate 2

3lim

2x

x

x

Solution Example 5 Calculate

0lim ( 1)ln

xx x

Solution

66

2.2 Continuity of Function Definition Function ( )f x is continuous at x a if all three following conditions are satisfied.

1. ( )f a exists 2. lim ( )

x af x

exists, i.e. lim ( )

x af x

lim ( )

x af x

3. lim ( )x a

f x

( )f a

Ramark 1. If at least one of the above conditions is not satisfied, then the given function is discontinuous at a . 2. If a function is continuous at point a , its graph has no break at point x a . Example 1 Let 2( ) 2 1f x x x Consider the continuity of this function at 0x : 1. (0) 1f exists 2.

0lim ( ) 1x

f x

exists, and

3. 0

lim (0)x

f

(0) 1f

67

Thus ( )f x is continuous at 0x as shown in Figure here.

-4 -3 -2 -1 1 2 3 4

-4

-2

2

4

x

y

Example 2 Let 21

, 1( ) 1

3 , 1

xx

f x x

x

Determine if this function is continuous at 1x . Solution

68

Example 3 Let 2 1 , 2

( ), 2

bx xf x

x x

Find b that makes this function continuous at 2x . Solution

69

Three Types of Discontinuities Consider continuity of ( )f x at a 1. Removable discontinuity It occurs when lim ( )

x af x

exists, but not equal to ( )f a or ( )f a is

undefined. For example, 1 , 0( )

, 0

xf x

x x

has a removable

discontinuity at 0x as show in Figure here.

2. Jump discontinuity or Ordinary discontinuity It occurs when lim ( )

x af x

does not exist due to unequal existence of

lim ( )x a

f x

and lim ( )x a

f x

. For example, 2 , 1

( ) 1 , 1

0 , 1

x

f x x

x

has

a jump discontinuity at 1, 1x .

1

-1

2

1

f(x) = x

f(0) = 1

70

3. Infinite discontinuity It occurs when either left limit or right limit or both limits does not

exist. For example, 2

1( )f x

x has an infinite discontinuity at 0x as

shown here. Algebraic properties of functions on continuity

1. If f and g are continuous at x a , then , . ,f

f g f gg

( ( ) 0g a )

and kf (k is a real number) are also continuous at x a 2. If f is continuous at x b and lim ( )

x ag x b

, then we have

lim ( )( ) ( )x a

fog x f b

3. If function g is continuous at x a and function f is continuous at ( )g a , then the composite function ( )fog is continuous at x a .

Example 4 Let 2

2

2( 4 2)( )

( 9)( 1)

x xf x

x x

Locate where this function is continuous.

71

Definition If function f is continuous everywhere in the interval ( , )a b , we say f is continuous in ( , )a b . Definition Function ( )f x is continuous in [ , ]a b where a b if 1. ( )f x is continuous in ( , )a b 2. lim ( ) ( )

x af x f a

3. lim ( ) ( )x b

f x f b

Example 5 Let 3( )

4

xg x

x

Locate where this function is continuous. Solution

72

Limit and Continuity Exercises 1. Find the limits of the following functions

(a) 3

( )1

xf x

x

Find

1lim ( )x

f x

(b) 2 2; 0

( )2 ; 0

x xg x

x x

Calculate

0lim ( )x

g x

(c) 26 3

lim2 1x

x

x

(d) 2 7

lim2 4x

x

x

2. Make the following functions continuous at x a

(a) 23

( )2

xf x

x , 0a

(b) 1( ) , , 1

1

nxg x n a

x

3. Locate domain that makes the following function continuous

(a) 2

2( )

3 28h x

x x

(b) 3( ) ( )( )k x x a x b

4. Find k that makes 2 5 6

; 2( ) 2

3; 2

x xx

f x x

kx x

continuous

everywhere. 5. Find k that makes each following limit exists

73

(a) 2

1

4lim

1x

x kx

x

(b) 4

2

3 5lim

2 1 kx

x x

x x

(c) 2 5

lim4

x

kxx

e

e

6. Compute the following limits

(a) 2

0

(3 ) 9limh

h

h

(b) 0

1/(1 ) 1limh

h

h

(c) 0

4 2limh

h

h

7. Compute the following limits

(a) 0

cos 1lim

sinx

x

x

(b) 0

sin3lim

sin5x

x

x

(c) lim sinx

xx

74

Answers to limit and continuity exercises 1. (a) (b) ‟2 (c) 3 (d) ‟1 / 2

2. (a) add 3(0)

2f

(b) add (1)g n 3. (a) 7,4x (b) ( , ) 4. 1 5. (a) 5 (b) greater than or equal to 4 (c) less than or equal to 2 6. (a) 6 (b) ‟1 (c) ‟1/16 7. (a) 0 (b) 3/5 (c)