Copyright © 2011 Pearson Education, Inc. Slide 12.1-1 Limits An Introduction To Limits Techniques for Calculating Limits One-Sided Limits; Limits Involving.

Copyright © 2011 Pearson Education, Inc. Slide 12.1-1

Limits

An Introduction To Limits

Techniques for Calculating Limits

One-Sided Limits; Limits Involving Infinity


Limit of a Function

The function

is not defined at x = 2, so its graph has a “hole” at x = 2.

2 4( )

2

xf x

x


Limit of a Function

Values of may be computed near x = 2

x approaches 2

f(x) approaches 4

2 4( )

2

xf x

x

x 1.9 1.99 1.999 2.001 2.01 2.1

f(x) 3.9 3.99 3.999 4.001 4.01 4.1


Limit of a Function

The values of f(x) get closer and closer to 4 as x gets closer and closer to 2.

We say that

“the limit of as x approaches 2 equals 4”

and write

2 4

2

x

x

2

2

4lim 4.

2x

x

x


Limit of a Function

Limit of a Function

Let f be a function and let a and L be real numbers. L is the limit of f(x) as x approaches a, written

if the following conditions are met.

1. As x assumes values closer and closer (but not equal ) to a on both sides of a, the corresponding values of f(x) get closer and closer (and are perhaps equal) to L.

2. The value of f(x) can be made as close to L as desired by taking values of x arbitrarily close to a.

lim ( ) ,x a

f x L


Finding the Limit of a Polynomial Function

Example Find

Solution The behavior of near x = 1 can be determined from a table of values,

x approaches 1

f(x) approaches 2

2

1lim ( 3 4).x

x x

x .9 .99 .999 1.001 1.01 1.1

f(x) 2.11 2.0101 2.001 1.999 1.9901 1.91

2( ) 3 4f x x x



Solution or from a graph of f(x).

We see that 2

1lim ( 3 4) 2.x

x x



Example Find where

Solution Create a graph and table.

3lim ( )x

f x

2 1 if 3( ) .

4 5 if 3

x xf x

x x



Solution x approaches 3

f(x) approaches 7

Therefore3

lim ( ) 7.x

f x

x 2.9 2.99 2.999 3.001 3.01 3.1

f(x) 6.8 6.98 6.998 7.004 7.04 7.4


Limits That Do Not Exist

• If there is no single value that is approached

by f(x) as x approaches a, we say that f(x)

does not have a limit as x approaches a,

or does not exist. 2

lim ( )x

f x


Determining Whether a Limit Exists

Example Find where

Solution Construct a table and graph

x 1.9 1.99 1.999 2.001 2.01 2.1

f(x) 2.6 2.96 2.996 1.003 1.03 1.3

2lim ( )x

f x

4 5 if 2( ) .

3 5 if 2

x xf x

x x



Solution

f(x) approaches 3 as x gets closer to 2 from the left,f(x) approaches 1 as x gets closer to 2 from the right.

Therefore, does not exist.2

lim ( )x

f x



Example Find where

Solution Construct a table and graph

0lim ( )x

f x 2

1( ) .f x

x

x -.1 -.01 -.001

f(x) 100 10,000 1,000,000

x .001 .01 .1

f(x) 1,000,000 10,000 100



Solution

As x approaches 0, the corresponding values of f(x) grow arbitrarily large.

Therefore, does not exist.20

1limx x


Limit of a Function

Conditions Under Which Fails To Exist

1. f(x) approaches a number L as x approaches a from the left and f(x) approaches a different number M as x approaches a from the right.

2. f(x) becomes infinitely large in absolute value as x approaches a from either side.

3. f(x) oscillates infinitely many times between two fixed values as x approaches a.

lim ( )x a

f x


Limits

1. An Introduction To Limits

2. Techniques for Calculating Limits

3. One-Sided Limits; Limits Involving Infinity


Techniques For Calculating Limits

Rules for Limits

1. Constant rule If k is a constant real number,

2. Limit of x rule For the following rules, we assume that and

both exist

3. Sum and difference rules

lim .x a

k k

lim .x a

x a

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

f x g x f x g x

lim ( )x a

f x

lim ( )x a

g x



Rules for Limits

4. Product Rule

5. Quotient Rule

provided

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

f x g x f x g x

lim ( )( )lim .

( ) lim ( )x a

x ax a

f xf x

g x g x

lim ( ) 0.x a

g x


Finding a Limit of a Linear Function

Example Find

Solution

Rules 1 and 4

Rules 1 and 2

4lim (3 2 ).x

x

4 4 4lim (3 2 ) lim 3 lim 2x x x

x x

4 43 lim 2 lim

x xx

3 2 4

11


Finding a Limit of a Polynomial Function with One Term

Example Find

Solution Rule 4

Rule 1

Rule 4

Rule 2

2

5lim 3 .x

x

2 2

5 5 5lim 3 lim 3 limx x x

x x

2

53 lim

xx

5 53 lim lim

x xx x

3 5 5

75


Finding a Limit of a Polynomial Function with One Term

For any polynomial function in the form ( ) ,nf x kx

lim ( ) ( ).n

x af x k a f a


Finding a Limit of a Polynomial Function

Example Find .

Solution

Rule 3

3

2lim (4 6 1)x

x x

3 3

2 2 2 2lim (4 6 1) lim 4 lim 6 lim 1x x x x

x x x x

34 2 6 2 1

21



Rules for Limits (Continued)

For the following rules, we assume that and

both exist.

6. Polynomial rule If p(x) defines a polynomial function, then

lim ( )x a

f x

lim ( )x a

g x

lim ( ) ( ).x a

p x p a




7. Rational function rule If f(x) defines a rational

function with then

8. Equal functions rule If f(x) = g(x) for all , then

lim ( ) ( ).x a

f x f a

( )

( )

p x

q x( ) 0q a

x a

lim ( ) lim ( ).x a x a

f x g x




9. Power rule For any real number k,

provided this limit exists.

lim[ ( )] lim ( )k

k

x a x af x f x




10. Exponent rule For any real number b > 0,

11. Logarithm rule For any real number b > 0 with ,

provided that

lim ( )( )lim .x af xf x

x ab b

1b

lim log ( ) log lim ( )b bx a x a

f x f x

lim ( ) 0.x a

f x


Finding a Limit of a Rational Function

Example Find

Solution Rule 7 cannot be applied directly since the denominator is 0. First factor the numerator and denominator

2

21

2 3lim .

3 2x

x x

x x

2

2

2 3 ( 3)( 1) 3

3 2 ( 2)( 1) 2

x x x x x

x x x x x



Solution Now apply Rule 8 with

and

so that f(x) = g(x) for all .

2

2

2 3( )

3 2

x xf x

x x

3

( )2

xg x

x

1x



Solution Rule 8

Rule 6

2

21 1

2 3 3lim lim

3 2 2x x

x x x

x x x

1 3

1 2

4


Limits

1 An Introduction To Limits

2 Techniques for Calculating Limits

3 One-Sided Limits; Limits Involving Infinity


One-Sided Limits

Limits of the form

are called two-sided limits since the values of x get close to a from both the right and left sides of a.

Limits which consider values of x on only oneside of a are called one-sided limits.

lim ( )x a

f x L


One-Sided Limits

The right-hand limit,

is read “the limit of f(x) as x approaches a from the right is L.”

As x gets closer and closer to a from the right (x > a), the values of f(x) get closer and closer to L.

lim ( )x a

f x L


One-Sided Limits

The left-hand limit,

is read “the limit of f(x) as x approaches a from the left is L.”

As x gets closer and closer to a from the right (x < a), the values of f(x) get closer and closer to L.

lim ( )x a

f x L


Finding One-Sided Limits of a Piecewise-Defined Function

Example Find and where

2lim ( )x

f x

2

6 if 2

5 if 2( )

1if 2

2

x x

xf x

x x

2lim ( )x

f x


Finding One-Sided Limits of a Piecewise-Defined Function

Solution Since x > 2 in use the formula

. In the limit , where x < 2, use

f(x) = x + 6.

2lim ( )x

f x

2lim ( )x

f x

2 2

2 2

2 2

1 1lim ( ) lim 2 2

2 2

lim ( ) lim ( 6) 2 6 8

x x

x x

f x x

f x x

21( )

2f x x


Infinity as a Limit

A function may increase without bound as x gets closer and closer to a from the right


Infinity as a Limit

The right-hand limit does not exist but the behavior is described by writing

If the values of f(x) decrease without bound, write

The notation is similar for left-handed limits.

lim ( )x a

f x

lim ( )x a

f x


Infinity as a Limit

Summary of infinite limits


Finding One-Sided Limits


Solution From the graph

2lim ( )x

f x

1( ) .

2f x

x

2lim ( )x

f x


Finding One-Sided Limits

Solution and the table

and2

lim ( )x

f x

2

lim ( ) .x

f x


Limits as x Approaches +

A function may approach an asymptotic value as

x moves in the positive or negative direction.

lim ( ) 2x

f x

lim ( ) 1x

g x



The notation,

is read “the limit of f(x) as x approaches infinity is L.”

The values of f(x) get closer and closer to L as x gets larger and larger.

lim ( )x

f x L



The notation,

is read “the limit of f(x) as x approaches negative infinity is L.”

The values of f(x) get closer and closer to L as x assumes negative values of larger and larger magnitude.

lim ( )x

f x L


Finding Limits at Infinity


Solution As the values of e-.25x get arbitrarily close to 0 so

lim ( )x

f x

.25

10( ) 5 .

1 xf x

e

lim ( )x

f x

x

10lim ( ) 5 15.

1 0xf x



Solution As the values of e-.25x get arbitrarily large so

x

lim ( ) 5 0 5.x

f x



Solution (Graphing calculator)



Limits at infinity of

For any positive real number n,

and1

lim 0nx x

1nx

1lim 0.

nx x


Finding a Limit at Infinity

Example Find

Solution Divide numerator and denominator by the highest power of x involved, x2.

2

2

5 7 1lim .

2 5x

x x

x x

2 2

2

2

7 155 7 1

lim lim1 52 5 2

x x

x x x xx x

x x



Solution 2 2

2

2

2

2

7 155 7 1

lim lim1 52 5 2

7 1lim 5

1 5lim 2

x x

x

x

x x x xx x

x x

x x

x x



Solution

2 2

2

2

1 1lim 5 7 lim lim5 7 1

lim1 12 5 lim 2 lim 5 lim

5 0 0 5

2 0 0 2

x x x

x

x x x

x x x xx x

x x

Copyright © 2011 Pearson Education, Inc. Slide 12.1-1 Limits An Introduction To Limits Techniques for Calculating Limits One-Sided Limits; Limits Involving.

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