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#1 2021-06-02 20:07:34

nycmathguy
Member
Registered: 2021-06-02
Posts: 53

Formal Definition of A Limit

The formal definition of a limit is like Chinese backwards to me.

1. What is the best way to learn the formal definition of a limit?

2. Can someone define the formal definition of a limit without using complicated
math jargon?

3. Can the formal definition of a limit be used to prove that the limit does exist for any function, including trigonometric functions?

Thank you

Last edited by nycmathguy (2021-06-02 20:40:05)

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#2 2021-06-02 22:37:48

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: Formal Definition of A Limit

hi nycmathguy

Welcome to the forum.

I've just logged in and see you have two posts on limits.  As this one came first I'll start by replying here.

If you're starting out on calculus I wouldn't get too hung up on the definition.  Despite what Stephen Hawking said, the universe hasn't got built in mathematical laws.  It's just what we (ie. centuries worth of mathematicians) have made up to suit our purposes.  I think the process which is usually described as differentiation from first principles is more important to grasp.  If you follow that then the world of calculus will open up to you.

If you have a straight line with points (3,7) and (5,15) lying on it, then we can calculate the gradient thus;

The theory surrounding y = mx + c shows us that this answer holds good if you take other points on the line.

But what if the function leads to a curve when you graph it.  A function like y = x^2 has a gradient that is continuously changing.  So what to do if we want the gradient at (say) (2,4)

You can take a point close to (2,4), let's say (2.5,6.25) and calculate the gradient of the chord that joins the points.

A rough sketch of the curve will show you that this answer is too big for the gradient at (2,4)

If you take a point to the left such as (1.5, 2.25) we can do the calculation again:

Our sketch shows us that this answer is too small for the gradient at (2,4).

So we already know that the answer we want is between 3.5 and 4.5.

You can move closer, and try (let's say) (2.1, 4.41):

and closer still:

It would be easy to just move our second point on top of the first:

o divided by o could be anything; what number do you have to multiply 0 by to get 0?

So that doesn't help.  But what we can observe is that our answers (4.5, 4.1, 4.01) are getting closer and closer to 4.  That's what we mean by a limit.  Is it true that the answers will keep getting closer and closer to 4. 
You could try (2.000001, whatever that is when squared).  Or use algebra:

Let's call the second point (x + Δ, x^2+ 2x.Δx + Δx^2) and the first (x,x^2)

Δx means a little bit in the x direction and it's a single symbol not two symbols multiplied together.

Now I'm going to let Δx get smaller and smaller.  We say it 'tends' to zero.  But if I do that with the above expression I'll get 0/0 again.  The 'trick' is to do some simplification first.

Now we can see why the answers with numbers got closer and closer to 4. That Δx means the answer is always above 4, but as it tends to zero, the gradient of the chord gets closer and closer to 2x.

That's the idea of a limit in practice.  I did it without worrying about the formal definition.  Much much later, if you get to University pure math courses the professors may start introducing the formal difinition just to make the subject 'properly rigorous', but you can get by without it.  Newton invented the process (at the same time as Leibnitz) and I'm pretty sure they just developed the method without too much formality.

There's more here:

https://www.mathsisfun.com/calculus/introduction.html

Hope that helps,

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2021-06-02 23:34:24

nycmathguy
Member
Registered: 2021-06-02
Posts: 53

Re: Formal Definition of A Limit

Bob wrote:

hi nycmathguy

Welcome to the forum.

I've just logged in and see you have two posts on limits.  As this one came first I'll start by replying here.

If you're starting out on calculus I wouldn't get too hung up on the definition.  Despite what Stephen Hawking said, the universe hasn't got built in mathematical laws.  It's just what we (ie. centuries worth of mathematicians) have made up to suit our purposes.  I think the process which is usually described as differentiation from first principles is more important to grasp.  If you follow that then the world of calculus will open up to you.

If you have a straight line with points (3,7) and (5,15) lying on it, then we can calculate the gradient thus;

The theory surrounding y = mx + c shows us that this answer holds good if you take other points on the line.

But what if the function leads to a curve when you graph it.  A function like y = x^2 has a gradient that is continuously changing.  So what to do if we want the gradient at (say) (2,4)

You can take a point close to (2,4), let's say (2.5,6.25) and calculate the gradient of the chord that joins the points.

A rough sketch of the curve will show you that this answer is too big for the gradient at (2,4)

If you take a point to the left such as (1.5, 2.25) we can do the calculation again:

Our sketch shows us that this answer is too small for the gradient at (2,4).

So we already know that the answer we want is between 3.5 and 4.5.

You can move closer, and try (let's say) (2.1, 4.41):

and closer still:

It would be easy to just move our second point on top of the first:

o divided by o could be anything; what number do you have to multiply 0 by to get 0?

So that doesn't help.  But what we can observe is that our answers (4.5, 4.1, 4.01) are getting closer and closer to 4.  That's what we mean by a limit.  Is it true that the answers will keep getting closer and closer to 4. 
You could try (2.000001, whatever that is when squared).  Or use algebra:

Let's call the second point (x + Δ, x^2+ 2x.Δx + Δx^2) and the first (x,x^2)

Δx means a little bit in the x direction and it's a single symbol not two symbols multiplied together.

Now I'm going to let Δx get smaller and smaller.  We say it 'tends' to zero.  But if I do that with the above expression I'll get 0/0 again.  The 'trick' is to do some simplification first.

Now we can see why the answers with numbers got closer and closer to 4. That Δx means the answer is always above 4, but as it tends to zero, the gradient of the chord gets closer and closer to 2x.

That's the idea of a limit in practice.  I did it without worrying about the formal definition.  Much much later, if you get to University pure math courses the professors may start introducing the formal difinition just to make the subject 'properly rigorous', but you can get by without it.  Newton invented the process (at the same time as Leibnitz) and I'm pretty sure they just developed the method without too much formality.

There's more here:

https://www.mathsisfun.com/calculus/introduction.html

Hope that helps,

Bob

Hi Bob. I'm happy to be a new member. It's good to know that there's no need to worry about the formal definition of a limit using delta and epsilon. When I saw this delta/epsilon stuff in chapter 1, I quickly became discouraged.

I think finding the limit of functions via graphs, substitution and algebraically is better for me at this level of learning. So, I will do as you say and skip all the rigorous, pure math jargon. This is great news. I can actually learn to find limits without pure mathematics, abstract material.

Thank you for taking the time to type all this information. Thank you for the link. I will visit the link to read and study more about limits in addition to the Michael Sullivan textbook. I will return with questions I get stuck with. Math work will be shown. It is my hope to be a regular user for years to come.

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#4 2021-06-03 10:33:12

nycmathguy
Member
Registered: 2021-06-02
Posts: 53

Re: Formal Definition of A Limit

Bob wrote:

hi nycmathguy

Welcome to the forum.

I've just logged in and see you have two posts on limits.  As this one came first I'll start by replying here.

If you're starting out on calculus I wouldn't get too hung up on the definition.  Despite what Stephen Hawking said, the universe hasn't got built in mathematical laws.  It's just what we (ie. centuries worth of mathematicians) have made up to suit our purposes.  I think the process which is usually described as differentiation from first principles is more important to grasp.  If you follow that then the world of calculus will open up to you.

If you have a straight line with points (3,7) and (5,15) lying on it, then we can calculate the gradient thus;

The theory surrounding y = mx + c shows us that this answer holds good if you take other points on the line.

But what if the function leads to a curve when you graph it.  A function like y = x^2 has a gradient that is continuously changing.  So what to do if we want the gradient at (say) (2,4)

You can take a point close to (2,4), let's say (2.5,6.25) and calculate the gradient of the chord that joins the points.

A rough sketch of the curve will show you that this answer is too big for the gradient at (2,4)

If you take a point to the left such as (1.5, 2.25) we can do the calculation again:

Our sketch shows us that this answer is too small for the gradient at (2,4).

So we already know that the answer we want is between 3.5 and 4.5.

You can move closer, and try (let's say) (2.1, 4.41):

and closer still:

It would be easy to just move our second point on top of the first:

o divided by o could be anything; what number do you have to multiply 0 by to get 0?

So that doesn't help.  But what we can observe is that our answers (4.5, 4.1, 4.01) are getting closer and closer to 4.  That's what we mean by a limit.  Is it true that the answers will keep getting closer and closer to 4. 
You could try (2.000001, whatever that is when squared).  Or use algebra:

Let's call the second point (x + Δ, x^2+ 2x.Δx + Δx^2) and the first (x,x^2)

Δx means a little bit in the x direction and it's a single symbol not two symbols multiplied together.

Now I'm going to let Δx get smaller and smaller.  We say it 'tends' to zero.  But if I do that with the above expression I'll get 0/0 again.  The 'trick' is to do some simplification first.

Now we can see why the answers with numbers got closer and closer to 4. That Δx means the answer is always above 4, but as it tends to zero, the gradient of the chord gets closer and closer to 2x.

That's the idea of a limit in practice.  I did it without worrying about the formal definition.  Much much later, if you get to University pure math courses the professors may start introducing the formal difinition just to make the subject 'properly rigorous', but you can get by without it.  Newton invented the process (at the same time as Leibnitz) and I'm pretty sure they just developed the method without too much formality.

There's more here:

https://www.mathsisfun.com/calculus/introduction.html

Hope that helps,

Bob

I found a tutor on You Tube who explained the delta/epsilon method to prove that a limit does exist. He made it seem easy. I actually understood what he said. I may be
tempted to try a few examples in the coming chapters but not so sure right now.
Name: MySecretMathTutor on You Tube. Check him out. Tell me what you think.

Thank you.

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