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The Formal Definition of a LimitNumeracy Workshop

Adrian Dudek

Adrian Dudek The Formal Definition of a Limit 2 / 37

Introduction

These slides cover the formal definition of a limit, and aim to be helpful for studentsstudying calculus to the level of MATH1001 or higher.

Drop-in Study Sessions: Monday, Wednesday, Thursday, 10am-12pm, Meeting Room2204, Second Floor, Social Sciences South Building, every week.

Website: Slides, notes, worksheets.

http://www.studysmarter.uwa.edu.au → Numeracy → Online Resources

Email: geoff.coates@uwa.edu.au

Adrian Dudek The Formal Definition of a Limit 3 / 37

The Fibonacci Sequence

The study of limits is, in a sense, the study of closeness.

Consider the famous Fibonacci sequence, defined by setting the first two terms of thesequence to 1, and furthermore defining each term to be the sum of the two before it.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .

Adrian Dudek The Formal Definition of a Limit 4 / 37

The Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .

Choose two terms in the Fibonacci sequence that sit next to each other, and divide thelarger by the smaller. For example, we choose 89 and 55, and then

89

55= 1.6181818 . . .

Let’s make a whole new sequence by taking the Fibonacci sequence and dividing eachterm by the term before it:

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, . . .

Adrian Dudek The Formal Definition of a Limit 5 / 37

The Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .

Choose two terms in the Fibonacci sequence that sit next to each other, and divide thelarger by the smaller. For example, we choose 89 and 55, and then

89

55= 1.6181818 . . .

Let’s make a whole new sequence by taking the Fibonacci sequence and dividing eachterm by the term before it:

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, . . .

Adrian Dudek The Formal Definition of a Limit 5 / 37

The Golden Ratio

1

1

2

1

3

2

5

3

8

5

13

8

21

13. . .

If we write each of these in their decimal notation to 4 decimal places we get:

1.0000 2.0000 1.5000 1.6667 1.6000 1.6250 1.6154 . . .

These numbers appear to be converging to a value.

In fact, this sequence approaches the golden number φ, which is equal to

1 +√

5

2= 1.618033 . . .

We say the limit of the sequence is φ.

Adrian Dudek The Formal Definition of a Limit 6 / 37

The Golden Ratio

1

1

2

1

3

2

5

3

8

5

13

8

21

13. . .

If we write each of these in their decimal notation to 4 decimal places we get:

1.0000 2.0000 1.5000 1.6667 1.6000 1.6250 1.6154 . . .

These numbers appear to be converging to a value.

In fact, this sequence approaches the golden number φ, which is equal to

1 +√

5

2= 1.618033 . . .

We say the limit of the sequence is φ.

Adrian Dudek The Formal Definition of a Limit 6 / 37

The Golden Ratio

1

1

2

1

3

2

5

3

8

5

13

8

21

13. . .

If we write each of these in their decimal notation to 4 decimal places we get:

1.0000 2.0000 1.5000 1.6667 1.6000 1.6250 1.6154 . . .

These numbers appear to be converging to a value.

In fact, this sequence approaches the golden number φ, which is equal to

1 +√

5

2= 1.618033 . . .

We say the limit of the sequence is φ.

Adrian Dudek The Formal Definition of a Limit 6 / 37

The Golden Ratio

1

1

2

1

3

2

5

3

8

5

13

8

21

13. . .

If we write each of these in their decimal notation to 4 decimal places we get:

1.0000 2.0000 1.5000 1.6667 1.6000 1.6250 1.6154 . . .

These numbers appear to be converging to a value.

In fact, this sequence approaches the golden number φ, which is equal to

1 +√

5

2= 1.618033 . . .

We say the limit of the sequence is φ.

Adrian Dudek The Formal Definition of a Limit 6 / 37

The Golden Ratio

1

1

2

1

3

2

5

3

8

5

13

8

21

13. . .

If we write each of these in their decimal notation to 4 decimal places we get:

1.0000 2.0000 1.5000 1.6667 1.6000 1.6250 1.6154 . . .

These numbers appear to be converging to a value.

In fact, this sequence approaches the golden number φ, which is equal to

1 +√

5

2= 1.618033 . . .

We say the limit of the sequence is φ.

Adrian Dudek The Formal Definition of a Limit 6 / 37

Limits: Example

Recall that a function takes in x-values and outputs y -values.

Sometimes we are more concerned with the value of a function near a point, rather thanat the point itself.

Usually this is because the function itself doesn’t exist at the point!

From this, the notion of a limit arises.

Adrian Dudek The Formal Definition of a Limit 7 / 37

Limits: Example

Recall that a function takes in x-values and outputs y -values.

Sometimes we are more concerned with the value of a function near a point, rather thanat the point itself.

Usually this is because the function itself doesn’t exist at the point!

From this, the notion of a limit arises.

Adrian Dudek The Formal Definition of a Limit 7 / 37

Limits: Example

Recall that a function takes in x-values and outputs y -values.

Sometimes we are more concerned with the value of a function near a point, rather thanat the point itself.

Usually this is because the function itself doesn’t exist at the point!

From this, the notion of a limit arises.

Adrian Dudek The Formal Definition of a Limit 7 / 37

Limits: Example

Recall that a function takes in x-values and outputs y -values.

Sometimes we are more concerned with the value of a function near a point, rather thanat the point itself.

Usually this is because the function itself doesn’t exist at the point!

From this, the notion of a limit arises.

Adrian Dudek The Formal Definition of a Limit 7 / 37

Limits: Example

Here is an example function f (x).

We can say that limx→−1 f (x) = −4.

Adrian Dudek The Formal Definition of a Limit 8 / 37

Limits: Example

Here is an example function f (x).

We can say that limx→−1 f (x) = −4.

Adrian Dudek The Formal Definition of a Limit 8 / 37

Limits: Example

The problem here is, we haven’t really justified this properly! We’ve just had a quickglance.

In fact, the function might be getting closer and closer to 4.000001, rather than 4.

So, we need a stronger argument for proving limits.

Let’s introduce our argument with an example.

Adrian Dudek The Formal Definition of a Limit 9 / 37

Limits: Example

The problem here is, we haven’t really justified this properly! We’ve just had a quickglance.

In fact, the function might be getting closer and closer to 4.000001, rather than 4.

So, we need a stronger argument for proving limits.

Let’s introduce our argument with an example.

Adrian Dudek The Formal Definition of a Limit 9 / 37

Limits: Example

The problem here is, we haven’t really justified this properly! We’ve just had a quickglance.

In fact, the function might be getting closer and closer to 4.000001, rather than 4.

So, we need a stronger argument for proving limits.

Let’s introduce our argument with an example.

Adrian Dudek The Formal Definition of a Limit 9 / 37

Limits: Example

Example: Prove that

limx→0

(x + 4) = 4.

What we are trying to show that the sum of 4 and something very close to 0, issomething very close to 4.

The idea is as follows.

You want to show that you can keep (x + 4) extremely close (as close as we like) to4 by keeping x extremely close to 0.

It’s this idea of closeness that requires formalising.

Adrian Dudek The Formal Definition of a Limit 10 / 37

Limits: Example

Example: Prove that

limx→0

(x + 4) = 4.

What we are trying to show that the sum of 4 and something very close to 0, issomething very close to 4.

The idea is as follows.

You want to show that you can keep (x + 4) extremely close (as close as we like) to4 by keeping x extremely close to 0.

It’s this idea of closeness that requires formalising.

Adrian Dudek The Formal Definition of a Limit 10 / 37

Limits: Example

Example: Prove that

limx→0

(x + 4) = 4.

What we are trying to show that the sum of 4 and something very close to 0, issomething very close to 4.

The idea is as follows.

You want to show that you can keep (x + 4) extremely close (as close as we like) to4 by keeping x extremely close to 0.

It’s this idea of closeness that requires formalising.

Adrian Dudek The Formal Definition of a Limit 10 / 37

Limits: Example

Example: Prove that

limx→0

(x + 4) = 4.

What we are trying to show that the sum of 4 and something very close to 0, issomething very close to 4.

The idea is as follows.

You want to show that you can keep (x + 4) extremely close (as close as we like) to4 by keeping x extremely close to 0.

It’s this idea of closeness that requires formalising.

Adrian Dudek The Formal Definition of a Limit 10 / 37

Limits: Example

You might be asked to keep 3.9 < x + 4 < 4.1. How would you do this?

Adding −4 to each part of the inequality gives −0.1 < x < 0.1.

You might be asked to keep 3.9999 < x + 4 < 4.0001. How would you do this?

Again, adding −4 to each part of the inequality gives −0.0001 < x < 0.0001.

Adrian Dudek The Formal Definition of a Limit 11 / 37

Limits: Example

You might be asked to keep 3.9 < x + 4 < 4.1. How would you do this?

Adding −4 to each part of the inequality gives −0.1 < x < 0.1.

You might be asked to keep 3.9999 < x + 4 < 4.0001. How would you do this?

Again, adding −4 to each part of the inequality gives −0.0001 < x < 0.0001.

Adrian Dudek The Formal Definition of a Limit 11 / 37

Limits: Example

You might be asked to keep 3.9 < x + 4 < 4.1. How would you do this?

Adding −4 to each part of the inequality gives −0.1 < x < 0.1.

You might be asked to keep 3.9999 < x + 4 < 4.0001. How would you do this?

Again, adding −4 to each part of the inequality gives −0.0001 < x < 0.0001.

Adrian Dudek The Formal Definition of a Limit 11 / 37

Limits: Example

You might be asked to keep 3.9 < x + 4 < 4.1. How would you do this?

Adding −4 to each part of the inequality gives −0.1 < x < 0.1.

You might be asked to keep 3.9999 < x + 4 < 4.0001. How would you do this?

Again, adding −4 to each part of the inequality gives −0.0001 < x < 0.0001.

Adrian Dudek The Formal Definition of a Limit 11 / 37

Limits: Example

Essentially, you are being asked how to keep 4− ε < x + 4 < 4 + ε, and you areanswering by saying that this is possible by −δ < x < δ.

However, there are infinitely many ε’s that they can throw at you, and you don’t want toplay this “choose δ” game forever! So you have to choose a δ that will beat any ε.

Adrian Dudek The Formal Definition of a Limit 12 / 37

Limits: Example

Essentially, you are being asked how to keep 4− ε < x + 4 < 4 + ε, and you areanswering by saying that this is possible by −δ < x < δ.

However, there are infinitely many ε’s that they can throw at you, and you don’t want toplay this “choose δ” game forever! So you have to choose a δ that will beat any ε.

Adrian Dudek The Formal Definition of a Limit 12 / 37

A Silly Game

A particularly silly game which two people can play is “who can say the highest number”.

This is a back-and-forth game just like the “choose delta” game which we just saw.

Every time your oppenent says a number, you can just say the number that is one higherthan their number to stay in the game!

However, rather than playing in this silly game, you could simply program a robot torespond for you with x + 1, where x is the number your opponent chooses.

We want to do the same thing with epsilons and deltas!

Adrian Dudek The Formal Definition of a Limit 13 / 37

A Silly Game

A particularly silly game which two people can play is “who can say the highest number”.

This is a back-and-forth game just like the “choose delta” game which we just saw.

Every time your oppenent says a number, you can just say the number that is one higherthan their number to stay in the game!

However, rather than playing in this silly game, you could simply program a robot torespond for you with x + 1, where x is the number your opponent chooses.

We want to do the same thing with epsilons and deltas!

Adrian Dudek The Formal Definition of a Limit 13 / 37

A Silly Game

A particularly silly game which two people can play is “who can say the highest number”.

This is a back-and-forth game just like the “choose delta” game which we just saw.

Every time your oppenent says a number, you can just say the number that is one higherthan their number to stay in the game!

However, rather than playing in this silly game, you could simply program a robot torespond for you with x + 1, where x is the number your opponent chooses.

We want to do the same thing with epsilons and deltas!

Adrian Dudek The Formal Definition of a Limit 13 / 37

A Silly Game

A particularly silly game which two people can play is “who can say the highest number”.

This is a back-and-forth game just like the “choose delta” game which we just saw.

Every time your oppenent says a number, you can just say the number that is one higherthan their number to stay in the game!

However, rather than playing in this silly game, you could simply program a robot torespond for you with x + 1, where x is the number your opponent chooses.

We want to do the same thing with epsilons and deltas!

Adrian Dudek The Formal Definition of a Limit 13 / 37

A Silly Game

A particularly silly game which two people can play is “who can say the highest number”.

This is a back-and-forth game just like the “choose delta” game which we just saw.

Every time your oppenent says a number, you can just say the number that is one higherthan their number to stay in the game!

However, rather than playing in this silly game, you could simply program a robot torespond for you with x + 1, where x is the number your opponent chooses.

We want to do the same thing with epsilons and deltas!

Adrian Dudek The Formal Definition of a Limit 13 / 37

Limits: Example

Let’s get back to our example!

So we wish to keep 4− ε < x + 4 < 4 + ε by keeping −δ < x < δ.

We simply do the same algebra we did to the specific examples: Add −4 to all sides of4− ε < x + 4 < 4 + ε to get:

−ε < x < ε

So in this case, ε = δ.

This is all we have to do to prove a limit! Provide a response δ in terms of any ε.

Of course, there is a little bit more to write out, but the hard work is done!

Adrian Dudek The Formal Definition of a Limit 14 / 37

Limits: Example

Let’s get back to our example!

So we wish to keep 4− ε < x + 4 < 4 + ε by keeping −δ < x < δ.

We simply do the same algebra we did to the specific examples: Add −4 to all sides of4− ε < x + 4 < 4 + ε to get:

−ε < x < ε

So in this case, ε = δ.

This is all we have to do to prove a limit! Provide a response δ in terms of any ε.

Of course, there is a little bit more to write out, but the hard work is done!

Adrian Dudek The Formal Definition of a Limit 14 / 37

Limits: Example

Let’s get back to our example!

So we wish to keep 4− ε < x + 4 < 4 + ε by keeping −δ < x < δ.

We simply do the same algebra we did to the specific examples: Add −4 to all sides of4− ε < x + 4 < 4 + ε to get:

−ε < x < ε

So in this case, ε = δ.

This is all we have to do to prove a limit! Provide a response δ in terms of any ε.

Of course, there is a little bit more to write out, but the hard work is done!

Adrian Dudek The Formal Definition of a Limit 14 / 37

Limits: Example

Let’s get back to our example!

So we wish to keep 4− ε < x + 4 < 4 + ε by keeping −δ < x < δ.

We simply do the same algebra we did to the specific examples: Add −4 to all sides of4− ε < x + 4 < 4 + ε to get:

−ε < x < ε

So in this case, ε = δ.

This is all we have to do to prove a limit! Provide a response δ in terms of any ε.

Of course, there is a little bit more to write out, but the hard work is done!

Adrian Dudek The Formal Definition of a Limit 14 / 37

Limits: Graphical Example

Suppose we have a function f (x), and we wish to show that limx→3 f (x) = 5.

Adrian Dudek The Formal Definition of a Limit 15 / 37

Limits: Graphical Example

Here ε = 1, and so we must choose a δ which works. We can see that δ = 1 is a finechoice.

Adrian Dudek The Formal Definition of a Limit 16 / 37

Limits: Graphical Example

Here ε = 1/2, and so we must choose a δ which works. We can see that δ = 1/2 is a finechoice.

Adrian Dudek The Formal Definition of a Limit 17 / 37

Limits: Graphical Example

Here ε = 0.25, and so we must choose a δ which works. We can see that δ = 0.25 is afine choice.

Adrian Dudek The Formal Definition of a Limit 18 / 37

Limits: Example

Example: Prove

limx→2

3x + 4 = 10

That is, show that as x gets really close to 2, then 3x + 4 gets really close to 10.

Adrian Dudek The Formal Definition of a Limit 19 / 37

Limits: Example

We want 3x + 4 to be really close to 10. We do this by specifying that the distancebetween them remain less than any positive number ε:

|3x + 4− 10| < ε

We want to show that this can be accomplished by keeping the distance between x and 2less than any amount δ:

|x − 2| < δ

The problem is solved by establishing an answer δ in terms of ε, so that you have ananswer for any ε they throw at you!

Adrian Dudek The Formal Definition of a Limit 20 / 37

Limits: Example

We want 3x + 4 to be really close to 10. We do this by specifying that the distancebetween them remain less than any positive number ε:

|3x + 4− 10| < ε

We want to show that this can be accomplished by keeping the distance between x and 2less than any amount δ:

|x − 2| < δ

The problem is solved by establishing an answer δ in terms of ε, so that you have ananswer for any ε they throw at you!

Adrian Dudek The Formal Definition of a Limit 20 / 37

Limits: Example

We want 3x + 4 to be really close to 10. We do this by specifying that the distancebetween them remain less than any positive number ε:

|3x + 4− 10| < ε

We want to show that this can be accomplished by keeping the distance between x and 2less than any amount δ:

|x − 2| < δ

The problem is solved by establishing an answer δ in terms of ε, so that you have ananswer for any ε they throw at you!

Adrian Dudek The Formal Definition of a Limit 20 / 37

Limits: Example

We usually proceed by rearranging the demand

|3x + 4− 10| < ε

into an inequality of the form |x − 2| < δ. Then we can simply read off what δ must be.

We start by writing the above without absolute value brackets.

−ε < 3x + 4− 10 < ε

Simplifying slightly we get

−ε < 3x − 6 < ε

Adrian Dudek The Formal Definition of a Limit 21 / 37

Limits: Example

We usually proceed by rearranging the demand

|3x + 4− 10| < ε

into an inequality of the form |x − 2| < δ. Then we can simply read off what δ must be.

We start by writing the above without absolute value brackets.

−ε < 3x + 4− 10 < ε

Simplifying slightly we get

−ε < 3x − 6 < ε

Adrian Dudek The Formal Definition of a Limit 21 / 37

Limits: Example

We usually proceed by rearranging the demand

|3x + 4− 10| < ε

into an inequality of the form |x − 2| < δ. Then we can simply read off what δ must be.

We start by writing the above without absolute value brackets.

−ε < 3x + 4− 10 < ε

Simplifying slightly we get

−ε < 3x − 6 < ε

Adrian Dudek The Formal Definition of a Limit 21 / 37

Limits: Example

We usually proceed by rearranging the demand

|3x + 4− 10| < ε

into an inequality of the form |x − 2| < δ. Then we can simply read off what δ must be.

We start by writing the above without absolute value brackets.

−ε < 3x + 4− 10 < ε

Simplifying slightly we get

−ε < 3x − 6 < ε

Adrian Dudek The Formal Definition of a Limit 21 / 37

Limits: Example

−ε < 3x − 6 < ε

Dividing by 3 we get

−ε/3 < x − 2 < ε/3

which is the same as

|x − 2| < ε/3

Thus, for any ε > 0 we choose, we would set δ = ε/3.

That is, by keeping |x − 2| < δ = ε/3, we guarantee that |3x + 4− 10| < ε.

Adrian Dudek The Formal Definition of a Limit 22 / 37

Limits: Example

−ε < 3x − 6 < ε

Dividing by 3 we get

−ε/3 < x − 2 < ε/3

which is the same as

|x − 2| < ε/3

Thus, for any ε > 0 we choose, we would set δ = ε/3.

That is, by keeping |x − 2| < δ = ε/3, we guarantee that |3x + 4− 10| < ε.

Adrian Dudek The Formal Definition of a Limit 22 / 37

Limits: Example

−ε < 3x − 6 < ε

Dividing by 3 we get

−ε/3 < x − 2 < ε/3

which is the same as

|x − 2| < ε/3

Thus, for any ε > 0 we choose, we would set δ = ε/3.

That is, by keeping |x − 2| < δ = ε/3, we guarantee that |3x + 4− 10| < ε.

Adrian Dudek The Formal Definition of a Limit 22 / 37

Limits: Example

−ε < 3x − 6 < ε

Dividing by 3 we get

−ε/3 < x − 2 < ε/3

which is the same as

|x − 2| < ε/3

Thus, for any ε > 0 we choose, we would set δ = ε/3.

That is, by keeping |x − 2| < δ = ε/3, we guarantee that |3x + 4− 10| < ε.

Adrian Dudek The Formal Definition of a Limit 22 / 37

Limits: Example

That is, by keeping |x − 2| < δ = ε/3, we guarantee that |3x + 4− 10| < ε.

We check as follows:

|3x + 4− 10| = |3x − 6|= 3|x − 2|< 3ε/3

< ε

Adrian Dudek The Formal Definition of a Limit 23 / 37

Limits: Example

That is, by keeping |x − 2| < δ = ε/3, we guarantee that |3x + 4− 10| < ε.

We check as follows:

|3x + 4− 10| = |3x − 6|

= 3|x − 2|< 3ε/3

< ε

Adrian Dudek The Formal Definition of a Limit 23 / 37

Limits: Example

That is, by keeping |x − 2| < δ = ε/3, we guarantee that |3x + 4− 10| < ε.

We check as follows:

|3x + 4− 10| = |3x − 6|= 3|x − 2|

< 3ε/3

< ε

Adrian Dudek The Formal Definition of a Limit 23 / 37

Limits: Example

That is, by keeping |x − 2| < δ = ε/3, we guarantee that |3x + 4− 10| < ε.

We check as follows:

|3x + 4− 10| = |3x − 6|= 3|x − 2|

< 3ε/3

< ε

Adrian Dudek The Formal Definition of a Limit 23 / 37

Limits: Example

That is, by keeping |x − 2| < δ = ε/3, we guarantee that |3x + 4− 10| < ε.

We check as follows:

|3x + 4− 10| = |3x − 6|= 3|x − 2|< 3ε/3

< ε

Adrian Dudek The Formal Definition of a Limit 23 / 37

Limits: Example

That is, by keeping |x − 2| < δ = ε/3, we guarantee that |3x + 4− 10| < ε.

We check as follows:

|3x + 4− 10| = |3x − 6|= 3|x − 2|< 3ε/3

< ε

Adrian Dudek The Formal Definition of a Limit 23 / 37

Limits: The Definition

In general, if we wish to show that

limx→a

f (x) = L

then we need to show that the distance between f (x) and L can be made as small as wewant, by making the distance between x and a sufficiently small.

That is, if somebody wants the distance between f (x) and L to be less than ε, then weneed to show that there is some δ (in terms of ε) where keeping the distance between x

and a less than δ guarantees this.

∀ε > 0 ∃ δ > 0 s.t |x − a| < δ ⇒ |f (x)− L| < ε

Adrian Dudek The Formal Definition of a Limit 24 / 37

Limits: The Definition

In general, if we wish to show that

limx→a

f (x) = L

then we need to show that the distance between f (x) and L can be made as small as wewant, by making the distance between x and a sufficiently small.

That is, if somebody wants the distance between f (x) and L to be less than ε, then weneed to show that there is some δ (in terms of ε) where keeping the distance between x

and a less than δ guarantees this.

∀ε > 0 ∃ δ > 0 s.t |x − a| < δ ⇒ |f (x)− L| < ε

Adrian Dudek The Formal Definition of a Limit 24 / 37

Limits: The Definition

In general, if we wish to show that

limx→a

f (x) = L

then we need to show that the distance between f (x) and L can be made as small as wewant, by making the distance between x and a sufficiently small.

That is, if somebody wants the distance between f (x) and L to be less than ε, then weneed to show that there is some δ (in terms of ε) where keeping the distance between x

and a less than δ guarantees this.

∀ε > 0 ∃ δ > 0 s.t |x − a| < δ ⇒ |f (x)− L| < ε

Adrian Dudek The Formal Definition of a Limit 24 / 37

Limits: The Definition

In general, if we wish to show that

limx→a

f (x) = L

then we need to show that the distance between f (x) and L can be made as small as wewant, by making the distance between x and a sufficiently small.

That is, if somebody wants the distance between f (x) and L to be less than ε, then weneed to show that there is some δ (in terms of ε) where keeping the distance between x

and a less than δ guarantees this.

∀ε > 0 ∃ δ > 0 s.t |x − a| < δ ⇒ |f (x)− L| < ε

Adrian Dudek The Formal Definition of a Limit 24 / 37

Limits: The Definition

Definition: We say that the limit of f (x) as x → a is L if

∀ε > 0 ∃ δ > 0 s.t |x − a| < δ ⇒ |f (x)− L| < ε

You need to remember this for tests and exams. Feel free to recite it at parties to testyour memory!

Adrian Dudek The Formal Definition of a Limit 25 / 37

Limits: The Definition

Definition: We say that the limit of f (x) as x → a is L if

∀ε > 0 ∃ δ > 0 s.t |x − a| < δ ⇒ |f (x)− L| < ε

You need to remember this for tests and exams. Feel free to recite it at parties to testyour memory!

Adrian Dudek The Formal Definition of a Limit 25 / 37

Infinite Limits

Sometimes we deal with limits as x → ±∞. One such example is:

limx→∞

(2 +

4

x

)= 2

This says, that as x gets really large, 2 +4

xgets really close to 2.

Once again, the way we prove this is the same!

You want to show that the distance between 2 +4

xand 2 can be made smaller than any

positive number ε, by making x larger than a corresponding number N.

Adrian Dudek The Formal Definition of a Limit 26 / 37

Infinite Limits

Sometimes we deal with limits as x → ±∞. One such example is:

limx→∞

(2 +

4

x

)= 2

This says, that as x gets really large, 2 +4

xgets really close to 2.

Once again, the way we prove this is the same!

You want to show that the distance between 2 +4

xand 2 can be made smaller than any

positive number ε, by making x larger than a corresponding number N.

Adrian Dudek The Formal Definition of a Limit 26 / 37

Infinite Limits: Example

Example: Prove that

limx→∞

(2 +

4

x

)= 2

We want to keep

∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε. How large does x need to be to guarantee this? That

is, find a number N, where keeping x > N will guarantee this.

Once again, we rearrange our original inequality for x .

Adrian Dudek The Formal Definition of a Limit 27 / 37

Infinite Limits: Example

Example: Prove that

limx→∞

(2 +

4

x

)= 2

We want to keep

∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε. How large does x need to be to guarantee this? That

is, find a number N, where keeping x > N will guarantee this.

Once again, we rearrange our original inequality for x .

Adrian Dudek The Formal Definition of a Limit 27 / 37

Infinite Limits: Example

Example: Prove that

limx→∞

(2 +

4

x

)= 2

We want to keep

∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε. How large does x need to be to guarantee this? That

is, find a number N, where keeping x > N will guarantee this.

Once again, we rearrange our original inequality for x .

Adrian Dudek The Formal Definition of a Limit 27 / 37

Infinite Limits: Example

We start with∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε

which gives

−ε < 4

x< ε

Multiplying through by x (keeping the inequality signs as they are, because x is positive)gives

−εx < 4 < εx

The right hand side of this says that 4 < εx . We rearrange this to get x >4

ε. So N =

4

ε.

Adrian Dudek The Formal Definition of a Limit 28 / 37

Infinite Limits: Example

We start with∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε

which gives

−ε < 4

x< ε

Multiplying through by x (keeping the inequality signs as they are, because x is positive)gives

−εx < 4 < εx

The right hand side of this says that 4 < εx . We rearrange this to get x >4

ε. So N =

4

ε.

Adrian Dudek The Formal Definition of a Limit 28 / 37

Infinite Limits: Example

We start with∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε

which gives

−ε < 4

x< ε

Multiplying through by x (keeping the inequality signs as they are, because x is positive)gives

−εx < 4 < εx

The right hand side of this says that 4 < εx . We rearrange this to get x >4

ε. So N =

4

ε.

Adrian Dudek The Formal Definition of a Limit 28 / 37

Infinite Limits: Example

We start with∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε

which gives

−ε < 4

x< ε

Multiplying through by x (keeping the inequality signs as they are, because x is positive)gives

−εx < 4 < εx

The right hand side of this says that 4 < εx . We rearrange this to get x >4

ε.

So N =4

ε.

Adrian Dudek The Formal Definition of a Limit 28 / 37

Infinite Limits: Example

We start with∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε

which gives

−ε < 4

x< ε

Multiplying through by x (keeping the inequality signs as they are, because x is positive)gives

−εx < 4 < εx

The right hand side of this says that 4 < εx . We rearrange this to get x >4

ε. So N =

4

ε.

Adrian Dudek The Formal Definition of a Limit 28 / 37

Infinite Limits: Example

So, we claim that

∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε whenever x > N =4

ε.

To finish off the proof nicely we will show the first statement is true under theassumption of the second:∣∣∣∣2 +

4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣The second statement re-arranges to give

4

x< ε.

Hence∣∣∣∣2 +4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣ < |ε| = ε (since ε > 0).

We have proven the limit.

Adrian Dudek The Formal Definition of a Limit 29 / 37

Infinite Limits: Example

So, we claim that

∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε whenever x > N =4

ε.

To finish off the proof nicely we will show the first statement is true under theassumption of the second:

∣∣∣∣2 +4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣The second statement re-arranges to give

4

x< ε.

Hence∣∣∣∣2 +4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣ < |ε| = ε (since ε > 0).

We have proven the limit.

Adrian Dudek The Formal Definition of a Limit 29 / 37

Infinite Limits: Example

So, we claim that

∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε whenever x > N =4

ε.

To finish off the proof nicely we will show the first statement is true under theassumption of the second:∣∣∣∣2 +

4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣

The second statement re-arranges to give

4

x< ε.

Hence∣∣∣∣2 +4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣ < |ε| = ε (since ε > 0).

We have proven the limit.

Adrian Dudek The Formal Definition of a Limit 29 / 37

Infinite Limits: Example

So, we claim that

∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε whenever x > N =4

ε.

To finish off the proof nicely we will show the first statement is true under theassumption of the second:∣∣∣∣2 +

4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣The second statement re-arranges to give

4

x< ε.

Hence∣∣∣∣2 +4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣ < |ε| = ε (since ε > 0).

We have proven the limit.

Adrian Dudek The Formal Definition of a Limit 29 / 37

Infinite Limits: Example

So, we claim that

∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε whenever x > N =4

ε.

To finish off the proof nicely we will show the first statement is true under theassumption of the second:∣∣∣∣2 +

4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣The second statement re-arranges to give

4

x< ε.

Hence∣∣∣∣2 +4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣

< |ε| = ε (since ε > 0).

We have proven the limit.

Adrian Dudek The Formal Definition of a Limit 29 / 37

Infinite Limits: Example

So, we claim that

∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε whenever x > N =4

ε.

To finish off the proof nicely we will show the first statement is true under theassumption of the second:∣∣∣∣2 +

4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣The second statement re-arranges to give

4

x< ε.

Hence∣∣∣∣2 +4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣ < |ε| = ε (since ε > 0).

We have proven the limit.

Adrian Dudek The Formal Definition of a Limit 29 / 37

Infinite Limits: Example

So, we claim that

∣∣∣∣2 +4

x− 2

∣∣∣∣ < ε whenever x > N =4

ε.

To finish off the proof nicely we will show the first statement is true under theassumption of the second:∣∣∣∣2 +

4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣The second statement re-arranges to give

4

x< ε.

Hence∣∣∣∣2 +4

x− 2

∣∣∣∣ =

∣∣∣∣ 4

x

∣∣∣∣ < |ε| = ε (since ε > 0).

We have proven the limit.

Adrian Dudek The Formal Definition of a Limit 29 / 37

Infinite Limits: The Definition

Definition: We say that the limit of f (x) as x →∞ is L if

∀ε > 0 ∃ N > 0 s.t x > N ⇒ |f (x)− L| < ε

Exercise: Try to write out the definition of a limit as x → −∞.

Adrian Dudek The Formal Definition of a Limit 30 / 37

Infinite Limits: The Definition

Definition: We say that the limit of f (x) as x →∞ is L if

∀ε > 0 ∃ N > 0 s.t x > N ⇒ |f (x)− L| < ε

Exercise: Try to write out the definition of a limit as x → −∞.

Adrian Dudek The Formal Definition of a Limit 30 / 37

Limit of a Sequence

What about limits of sequences?

We might want to show that the sequence 1, 1/2, 1/3, 1/4 . . . converges to 0.

Adrian Dudek The Formal Definition of a Limit 31 / 37

Limit of a Sequence

The limit of a sequence requires the same sort of approach as for infinite limits.

If we believe a sequence an → L, then we must show that we can keep an arbitrarily closeto L, by starting our sequence far enough to the right.

Starting our sequence far enough to the right means that the index n of our sequence ancommences after some positive number N.

Adrian Dudek The Formal Definition of a Limit 32 / 37

Limit of a Sequence

The limit of a sequence requires the same sort of approach as for infinite limits.

If we believe a sequence an → L, then we must show that we can keep an arbitrarily closeto L, by starting our sequence far enough to the right.

Starting our sequence far enough to the right means that the index n of our sequence ancommences after some positive number N.

Adrian Dudek The Formal Definition of a Limit 32 / 37

Limit of a Sequence

The limit of a sequence requires the same sort of approach as for infinite limits.

If we believe a sequence an → L, then we must show that we can keep an arbitrarily closeto L, by starting our sequence far enough to the right.

Starting our sequence far enough to the right means that the index n of our sequence ancommences after some positive number N.

Adrian Dudek The Formal Definition of a Limit 32 / 37

Limit of a Sequence: Example

Let (an)n≥1 be the sequence defined by an = 1/(1 + n). Show that

limn→∞

an = 0

We want to show that |1/(1 + n)− 0| < ε whenever we keep n > N for some positivenumber N.

This will work just as before! We want to find N in terms of ε.

Adrian Dudek The Formal Definition of a Limit 33 / 37

Limit of a Sequence: Example

Let (an)n≥1 be the sequence defined by an = 1/(1 + n). Show that

limn→∞

an = 0

We want to show that |1/(1 + n)− 0| < ε whenever we keep n > N for some positivenumber N.

This will work just as before! We want to find N in terms of ε.

Adrian Dudek The Formal Definition of a Limit 33 / 37

Limit of a Sequence: Example

Let (an)n≥1 be the sequence defined by an = 1/(1 + n). Show that

limn→∞

an = 0

We want to show that |1/(1 + n)− 0| < ε whenever we keep n > N for some positivenumber N.

This will work just as before! We want to find N in terms of ε.

Adrian Dudek The Formal Definition of a Limit 33 / 37

Limit of a Sequence: Example

We start with

|1/(1 + n)− 0| < ε

which gives

−ε < 1/(1 + n) < ε.

Multiplying through by (1 + n) gives:

−ε(1 + n) < 1 < ε(1 + n)

Dividing through by ε gives:

−(1 + n) < 1/ε < 1 + n

Adrian Dudek The Formal Definition of a Limit 34 / 37

Limit of a Sequence: Example

We start with

|1/(1 + n)− 0| < ε

which gives

−ε < 1/(1 + n) < ε.

Multiplying through by (1 + n) gives:

−ε(1 + n) < 1 < ε(1 + n)

Dividing through by ε gives:

−(1 + n) < 1/ε < 1 + n

Adrian Dudek The Formal Definition of a Limit 34 / 37

Limit of a Sequence: Example

We start with

|1/(1 + n)− 0| < ε

which gives

−ε < 1/(1 + n) < ε.

Multiplying through by (1 + n) gives:

−ε(1 + n) < 1 < ε(1 + n)

Dividing through by ε gives:

−(1 + n) < 1/ε < 1 + n

Adrian Dudek The Formal Definition of a Limit 34 / 37

Limit of a Sequence: Example

We start with

|1/(1 + n)− 0| < ε

which gives

−ε < 1/(1 + n) < ε.

Multiplying through by (1 + n) gives:

−ε(1 + n) < 1 < ε(1 + n)

Dividing through by ε gives:

−(1 + n) < 1/ε < 1 + n

Adrian Dudek The Formal Definition of a Limit 34 / 37

Limit of a Sequence: Example

We start with

|1/(1 + n)− 0| < ε

which gives

−ε < 1/(1 + n) < ε.

Multiplying through by (1 + n) gives:

−ε(1 + n) < 1 < ε(1 + n)

Dividing through by ε gives:

−(1 + n) < 1/ε < 1 + n

Adrian Dudek The Formal Definition of a Limit 34 / 37

Limit of a Sequence: Example

−(1 + n) < 1/ε < 1 + n

Now rearranging the rightmost part of this inequality gives us n > 1/ε− 1.

So, to keep |1/(1 + n)− 0| < ε, we need to start the sequence off after N = 1/ε− 1. Weshow that this works as follows:

| 1

1 + n− 0| =

1

1 + n<

1

1 + 1/ε− 1=

1

1/ε= ε

and we are done.

Adrian Dudek The Formal Definition of a Limit 35 / 37

Limit of a Sequence: Example

−(1 + n) < 1/ε < 1 + n

Now rearranging the rightmost part of this inequality gives us n > 1/ε− 1.

So, to keep |1/(1 + n)− 0| < ε, we need to start the sequence off after N = 1/ε− 1. Weshow that this works as follows:

| 1

1 + n− 0| =

1

1 + n<

1

1 + 1/ε− 1=

1

1/ε= ε

and we are done.

Adrian Dudek The Formal Definition of a Limit 35 / 37

Limit of a Sequence: Example

−(1 + n) < 1/ε < 1 + n

Now rearranging the rightmost part of this inequality gives us n > 1/ε− 1.

So, to keep |1/(1 + n)− 0| < ε, we need to start the sequence off after N = 1/ε− 1. Weshow that this works as follows:

| 1

1 + n− 0| =

1

1 + n<

1

1 + 1/ε− 1=

1

1/ε= ε

and we are done.

Adrian Dudek The Formal Definition of a Limit 35 / 37

Limit of a Sequence: The Definition

Definition: We say that the limit of an as n→∞ is L if

∀ε > 0 ∃ N > 0 s.t n > N ⇒ |an − L| < ε

We often say an → L, rather than limn→∞ an = L.

Adrian Dudek The Formal Definition of a Limit 36 / 37

Limit of a Sequence: The Definition

Definition: We say that the limit of an as n→∞ is L if

∀ε > 0 ∃ N > 0 s.t n > N ⇒ |an − L| < ε

We often say an → L, rather than limn→∞ an = L.

Adrian Dudek The Formal Definition of a Limit 36 / 37

Using STUDYSmarter Resources

This resource was developed for UWA students by the STUDYSmarter team for thenumeracy program. When using our resources, please retain them in their original form

with both the STUDYSmarter heading and the UWA crest.

Adrian Dudek The Formal Definition of a Limit 37 / 37