12-3 – Infinite Sequences and Series · 05-08-2016  · 13.4 Limits of Infinite Sequences 1 limsin . nof n §· ¨¸ ©¹ As n gets larger, 1 𝑛 gets smaller and smaller. s s,

13.4 Limits of Infinite Sequences

Objective

To find or estimate the limit of an infinite sequence

or to determine that the limit does not exist.

• Sequences with a numerical limit

• Sequences with no limit

• Sequences with a limit of infinity

Limits

Limits are written in the form below.

It is read “The limit of 1 over n as n

approaches infinity”. 1limn n

Limits are used to determine how a function,

sequence, or series will behave as the

independent variable approaches a certain

value, often infinity.

A graph of the sequence 1

𝑛for 𝑛 = 1,2,3,… , 10

1limn n

The terms are 1

1,1

2,1

3, … ,

1

10

= 0

2,4,8,16,32,…… . . lim𝑛→∞2𝑛 = ∞

13.4 Limits of Infinite Sequences

−1,1,−1,1,−1…… . .

lim𝑛→∞

−5𝑛 =−∞

−5,−10,−15,−20,…… . .

lim𝑛→∞

(−1)𝑛 = 𝐷𝑁𝐸 (does not exist)

13.4 Limits of Infinite Sequences

Situations in Which a Sequence Has No Limit

If the terms of a sequence do not " "

on a single value, we say that the limit of the

sequence

home in

DNE.

1

11 2 3 4, , , ,...., ,...

2 3 4 5 1

nn

n

13.4 Limits of Infinite Sequences

1

11 2 3 4, , , ,...., ,...

2 3 4 5 1

nn

n

1

1lim

1

n

n

nDNE

n

13.4 Limits of Infinite Sequences

Infinite Limits

increaseWhen the terms of a sequence or

without decre bouase nd.

3,7,11,15,....,4 1,...n lim 4 1n

n

-10, 100, 1000,...., 10 ,...n lim 10nn

13.4 Limits of Infinite Sequences

11 1 1 11 ,1 ,1 ,1 ,....1 ,...

1 2 3 4

n

n

11 1lim

n

n n

Numerically investigate

13.4 Limits of Infinite Sequences1

limsin .n n

As n gets larger, 1

𝑛gets smaller and smaller.

1

1,1

2,1

3,1

4,1

5,… . .

In fact, 1

𝑛approaches 0. Therefore, lim

𝑛→∞

1

𝑛= 0.

It follows that lim𝑛→∞

𝑠𝑖𝑛1

𝑛=𝑠𝑖𝑛0 = 0

Numerically investigate

13.4 Limits of Infinite Sequences

1limsin .n n

1limsin 0n n

1 1 1 1 1, , , ,.... ,...

2 4 8 16 2

n

10

2lim

n

n

nnt

1/ 21/ 4

1/81/16

1234

Numerically investigate

13.4 Limits of Infinite Sequences

0.99limn

n

Numerically investigate

13.4 Limits of Infinite Sequences

0.99limn

n

Numerically investigate

13.4 Limits of Infinite Sequences

0.99limn

n

Numerically investigate

13.4 Limits of Infinite Sequences

0.99limn

n

0.99 0limn

n

13.4 Limits of Infinite Sequences

If 1, then li

Theo em

m 0

r

n

nr r

0.99limn

n

0

Rational Function Reminders

Rational functions have the form:𝑓(𝑥)

𝑔(𝑥)

where f and g are polynomial functions.

The degree of a polynomial is equal to the

largest exponent in the equation when

written in standard form.

3𝑥4 + 𝑥2 − 1

2𝑥3 − 𝑥2

Guidelines for Finding Limits at ∞

of Rational Functions1. If the degree of the numerator is ___________ the degree

of the denominator, then the limit of the rational function

is _______.

2. If the degree of the numerator is ___________ the degree

of the denominator, then the limit of the rational function

is __.

3. If the degree of the numerator is _______ the degree of

the denominator, then the limit of the rational function is

the __________________ _______________________.

greater than

0

less than

equal to

ratio of the leading coefficients

infinite

4 3

4

3 1lim

2 7nn n

n n

=

1

2

3

3

6lim

3nn n

n

=1

3

Find the limit for each sequence. You can check

by graphing.

lim𝑛→∞

6𝑛5 + 4𝑛3

2𝑛2 − 𝑛 = ∞

=1

2

= 0

= ∞

PracticeEvaluate the following.

3

4 2limn

n

n

2

2

3 2 5lim

4 1

n

n n

n

lim𝑛→∞

1.0001𝑛

lim𝑛→∞

𝑐𝑜𝑠1

𝑛lim𝑛→∞

𝑙𝑜𝑔 𝑐𝑜𝑠1

𝑛

3

40 ∞

1 03

4 2

2 3lim

3x

x x

x x x

0

=2

5

3 2

5 6lim

2 8x

x x

x x

∞8

8 2

6 12 17lim

18 13 24x

x x

x x

1

3lim𝑥→∞

4𝑥3 + 2𝑥2 − 5

𝑥3 + 4

Infinite Series Video

https://www.youtube.com/watch?v=jktaz0ZautY

Homework

Page 496 #1-29

odds