Chapter 2Chapter 2Limit and continuityLimit and continuity

Tangent lines and length of the curveTangent lines and length of the curve The concept of limitThe concept of limit Computation of limitComputation of limit ContinuityContinuity Limit involving infinity (asymptotesLimit involving infinity (asymptotes

1)Tangent lines and the length 1)Tangent lines and the length of the curveof the curve

Tangent lineTangent line

Lets zoom the graphLets zoom the graph

•Now we can see that the slope get closerNow we can see that the slope get closer

To get the correct tangent line of the graph at any point, we To get the correct tangent line of the graph at any point, we need to zoom the graph as can as possibleneed to zoom the graph as can as possible

The length of curveThe length of curve

2 2

2 2

(4 1) (3 0)

+ (16 1) (6 3)

18 234 19.53

d

2 22

2 2

2 2

(0 4) (2 0)

(4 0) (4 2)

(16 4) (6 4)

21,1

d

2 23

2 2

2 2

2 2

2 2

(1 4) (1 0)

+ (0 1) (2 1)

+ (2 0) (3.4 2)

+ (6 2) (4.4 3.4)

+ (16 6) (6 4.4)

21.25

d

2)The concept of limit2)The concept of limit

Example 1Example 1

f is not defined at x=2f is not defined at x=2

Example 2Example 2

g is not defined at x=2g is not defined at x=2

2( 4)( )

( 2)

xf x

x

2( 5)( )

( 2)

xg x

x

Thus, we can conclude that the limit of f(x) when x approaches 2 is 4 and we write Thus, we can conclude that the limit of f(x) when x approaches 2 is 4 and we write

We can see that when x get closer We can see that when x get closer to 2 from left side, f(x) get closer to to 2 from left side, f(x) get closer to 44

1.91.9 3.93.9

1.991.99 3.993.99

1.9991.999 3.9993.999

1.99991.9999 3.99993.9999

2( 4)( )

( 2)

xf x

x

x2( 4)

( )( 2)

xf x

x

2.12.1 4.14.1

2.012.01 4.014.01

2.0012.001 4.0014.001

2.00012.0001 4.00014.0001

x2( 4)

( )( 2)

xf x

x

We can see also that when x get closer We can see also that when x get closer to 2 from right side, f(x) get closer to 4to 2 from right side, f(x) get closer to 4

2lim ( ) 4x

f x

Since the limits from the two sides are different, we conclude that the limit of g(x) Since the limits from the two sides are different, we conclude that the limit of g(x) when x approaches 2 doesn’t exist when x approaches 2 doesn’t exist

We can see that when x get closer We can see that when x get closer to 2 from left side, g(x) increase to 2 from left side, g(x) increase very fast toward very fast toward

1.91.9 13.913.9

1.991.99 103.99103.99

1.9991.999 1003.9991003.999

1.99991.9999 10003.999910003.9999

2( 5)( )

( 2)

xg x

x

x2( 5)

( )( 2)

xg x

x

2.12.1 -5.9-5.9

2.012.01 -95.99-95.99

2.0012.001 -995.999-995.999

2.00012.0001 -9995.9999-9995.9999

x

2( 5)( )

( 2)

xg x

x

We can see also that when x get closer We can see also that when x get closer to 2 from right side,g(x) decrease very to 2 from right side,g(x) decrease very fast toward fast toward

2lim ( ) does not existx

g x

Def:Def: A limit exist if and only if the two one-sided limit exist and are equal A limit exist if and only if the two one-sided limit exist and are equal for some L, if and only if for some L, if and only if

Exercise:Exercise: Find out why existFind out why exist

while does not exist while does not exist

Example:Example: Evaluate the below limit is it existEvaluate the below limit is it exist

lim ( ) x af x L lim ( ) lim ( )

x a x af x f x L

22

2lim

4x

x

x

22

2lim

4x

x

x

0

| |limx

x

x

0 0

0 0

0 0 0

| |lim lim 1

| |lim lim 1

| | | | | |lim lim lim does not exist

x x

x x

x x x

x x

x xx x

x xx x x

x x x

3)Computation of the limit3)Computation of the limit

Theorem1:Theorem1:

ExampleExample evaluate the following evaluate the following limitslimits

suppose that lim ( ) and lim ( ) both exist and

let c be any constant, the following then apply

1) lim ( . ( )) . lim ( )

2) lim ( ( ) ( )) lim ( ) lim ( )

3) lim ( ( ). ( )) lim (

x a x a

x a x a

x a x a x a

x a x a

f x f x

c f x c f x

f x g x f x g x

f x g x f x

). lim ( ),

lim ( )( )4) lim ( ) /( lim ( ) 0)

( ) lim ( )

x a

x a

x a x ax a

g x and

f xf xg x

g x g x

2

20

2

2

(2 )( 1)1) lim

2 3

42) lim

2

x

x

x x

x x

x

x

Theorem2:Theorem2:

For any Polynomial p(x) and any For any Polynomial p(x) and any real number a, real number a,

Theorem 3:Theorem 3:

suppose that , thensuppose that , then

Example:Example: evaluate evaluate

lim ( ) ( )x a

p x p a

lim ( )x a

f x L

lim ( ) lim ( ) nn nx a x a

f x f x L

5

2

27

2

1) lim 2 2

22) lim

2

x

x

x

x

x

Theorem 4:Theorem 4:

Example:Example: evaluate the following evaluate the following limitlimit

Theorem 5Theorem 5 (squeeze theorem)(squeeze theorem) Suppose thatSuppose that

for all x in some interval for all x in some interval and and

for some number L, then alsofor some number L, then also

Example:Example: evaluate the limitevaluate the limit

ExampleExample: ( a limit of piecewise-: ( a limit of piecewise-defined function)defined function)

1 1

1 1

1 1

For any real number , we have

1) lim sin sin

2) lim cos cos

3) lim

4) lim ln ln , for 0

5) lim sin sin for 1 1

6) lim cos cos for 1 1

7) lim tan tan for

8

x a

x a

x a

x a

x a

x a

x a

x a

a

x a

x a

e e

x a a

x a a

x a a

x a a

( ))if p(x) is Ploy and lim ( )

lim ( ( ))

x p a

x a

f x L

f p x L

1

1/2

0

1) lim sin

2) lim .cotx

x

x

x x

( ) ( ) ( )h x f x g x

( , )c b

lim ( ) lim ( ) / ( , )x a x a

h x g x L a c d

lim ( )x a

f x L

2

0lim cos (1/ )x

x x

2

0

2cos 1 for 0lim ( ), where ( )

4 for 0xx

x x xf x f x

e x

4)Continuity of Functions4)Continuity of Functions

Definition Definition

A function f is continuous at a point A function f is continuous at a point xx = = a a if if

is defined is defined 1) ( )f a

2) lim ( ) exist and

3) lim ( ) ( )x a

x a

f x

f x f a

So the function sin(x) is continuous at x=0

So the function f(x) is not continuous at x=3

So the function f(x) is not continuous at x=0

Example Where the function tan is continuous?y x

Solution

sinThe function tan is continuous whenever cos 0.

cos

Hence tan is continuous at , .2

xy x x

x

y x x n n

example 2

1Where the function f sin is continuous?

1

Solution 2

1The function f sin is continuous at all points

1

where it takes finite values.

2 2

1 1If 1, is not finite, and sin is undefined.

1 1

2 2

1 1If 1, is finite, and sin is defined and also finite.

1 1

2

1Hence sin is continuous for 1.

1

0

0

A number for which an expression f either is undefined or

infinite is called a of the function f . The singularity is

said to be , if f can be defi

singularity

removab ned in such a way le that

x x

x

0

the function f becomes continous at .x x

Problem 14

2

0

0

0

Which of the following functions have removable

singularities at the indicated points?

2 8a) f , 2

21

b) g , 11

1c) h sin , 0

x xx x

xx

x xx

t t tt

Answer

Removable

Removable

Not removable

Theorem 1Theorem 1 All PolynomialAll Polynomial

are continuous everywhere, is are continuous everywhere, is continuous for all x if n is odd, andcontinuous for all x if n is odd, and

continuous for all x 0 if n is continuous for all x 0 if n is even. Also is continuous for all even. Also is continuous for all , and , and for for

Theorem 3Theorem 3 Suppose that and Suppose that and

is continuous at , then is continuous at , then

1sin ,cos , tan , xx x x e

n x

ln x

0x 1 1sin andcos x x 1 1x

Theorem 2Theorem 2 Suppose that f(x) and g(x) are Suppose that f(x) and g(x) are

continuous at x=a, thencontinuous at x=a, then (f+g)(x) is continuous at x=a(f+g)(x) is continuous at x=a (f-g)(x) is continuous at x=a(f-g)(x) is continuous at x=a (f.g)(x) is continuous at x=a(f.g)(x) is continuous at x=a (f/g)(x) is continuous at x=a if (f/g)(x) is continuous at x=a if

g(a) 0g(a) 0

lim ( )x a

g x L

f

L

lim ( ( )) (lim ( )) ( )x a x a

f g x f g x f L

ExampleExample Determine where Determine where

is continuous? is continuous?

2( ) cos( 3)f x x x

Corollary Corollary Suppose that is continuous Suppose that is continuous

at and is at and is continuous at , thencontinuous at , then

is continuous at is continuous at

( )g x a

( )f x ( )g x

( )fog x a

Practice on continuityPractice on continuity Exercise 11Exercise 11: explain why each : explain why each

function is discontinuous at the given function is discontinuous at the given point by indicating which of the three point by indicating which of the three condition in definition are not metcondition in definition are not met

2 if 2

( ) 3 if 2

3 2 if 2

x x

f x x

x x

Exercise 23,19Exercise 23,19:: find all discontinuity find all discontinuity of f(x). If the discontinuity is of f(x). If the discontinuity is removable, introduce the new removable, introduce the new function that remove the function that remove the discontinuity:discontinuity:

2

3

2

3 1 if 1

1) ( ) 5 if 1 1

3 if 1

2) ( ) ln( )

x x

f x x x x

x x

f x x x

Exercise 34Exercise 34: determine the value of : determine the value of a and b that make the given a and b that make the given function continuous function continuous

1

2

1 if 0

( ) sin (x/2) if 0 2

if 2

xae x

f x x

x x b x

Theorem4Theorem4: (intermediate value : (intermediate value theorem)theorem)

Suppose that is continuous on Suppose that is continuous on closed interval [a, b], and W is any closed interval [a, b], and W is any number between f(a) and f(b). number between f(a) and f(b). Then, there is a numberThen, there is a number

For which For which

( )f x[ , ]c a b

( )f c W

corollary2corollary2: Suppose that f(x) : Suppose that f(x) is continuous on closed is continuous on closed interval (a, b), and f(a) and interval (a, b), and f(a) and f(b) have opposite signs f(b) have opposite signs (f(a).f(b)<0), so there is at (f(a).f(b)<0), so there is at least one number least one number

for which f(c)=0for which f(c)=0( , )c a b

Limit involving infinity, Limit involving infinity, Asymptotes Asymptotes Example 1: Example 1:

0

1limx x