Limit Identities

[holysword]

I'm asking it here because I don't even know HOW to ask this!

Basically, I'm looking for some identities regarding the "tends to" part of the limit. So for instance:

[latex]\lim_{x\rightarrow b} f(x) = \lim_{x\rightarrow 0} f(x+b)[/latex]

Is the above expression correct? If so, is it always correct, at least for Hilbert spaces? What else can you do with the "tends to" part, can you rename variables there? References are welcome.

[Sriman Dutta]

It looks correct. Let's try a function.

[math] f(x)=3x^2-a[/math] , a is a constant.

 

 

[math] \lim_{x\rightarrow 0} f(x+a)= 3(0+a)^2 - a=3a^2-a=\lim_{x\rightarrow a} f(x)[/math]

Edited May 10, 2017 by Sriman Dutta

[uncool]

It is correct; the interesting part comes with the question of why.

 

Can you think of both a conceptual (dealing with the concept of limits) and a formal (dealin with epsilon-delta definition) reason why?

[holysword]

It looks correct. Let's try a function.

[math] f(x)=3x^2-a[/math] , a is a constant.

 

 

[math] \lim_{x\rightarrow 0} f(x+a)= 3(0+a)^2 - a=3a^2-a=\lim_{x\rightarrow a} f(x)[/math]

I don't think that examples are a safe way to prove things.

 

It is correct; the interesting part comes with the question of why.

 

Can you think of both a conceptual (dealing with the concept of limits) and a formal (dealin with epsilon-delta definition) reason why?

No, I cannot. I'm not a maths student. Can you enlighten me please?

[Sriman Dutta]

I don't think that examples are a safe way to prove things.

 

Err, you're correct. That's not a good way to prove an identity. But that little example showed correct results with that.

[studiot]

 

I'm asking it here because I don't even know HOW to ask this!

 

Basically, I'm looking for some identities regarding the "tends to" part of the limit. So for instance:

[latex]\lim_{x\rightarrow b} f(x) = \lim_{x\rightarrow 0} f(x+b)[/latex]

Is the above expression correct? If so, is it always correct, at least for Hilbert spaces? What else can you do with the "tends to" part, can you rename variables there? References are welcome.

 

 

You say you are not a maths student and I'm sure uncool can put you right on the maths.

 

But I am a little concerned by the phrase I'm looking for some identities regarding the 'tends to' part of the limit.

 

A limit is a process and the result of that process.

You need both parts of the definition to understand limits.

 

This is simlar to the comment that a function is a process and the result of that process and also includes the 'working material' of the function, called the domain.

 

Hilbert spaces are rather esoteric and advanced for someone who is not a maths student, if you would like to explain you interest it would help couch replies in suitable language.

[holysword]

 

You say you are not a maths student and I'm sure uncool can put you right on the maths.

 

But I am a little concerned by the phrase I'm looking for some identities regarding the 'tends to' part of the limit.

 

A limit is a process and the result of that process.

You need both parts of the definition to understand limits.

 

This is simlar to the comment that a function is a process and the result of that process and also includes the 'working material' of the function, called the domain.

 

Hilbert spaces are rather esoteric and advanced for someone who is not a maths student, if you would like to explain you interest it would help couch replies in suitable language.

 

If anything, Hilbert spaces are "well-behaved" - that's why I asked targeting those spaces instead of something more complicated. But if you want to simplify, we could go with finite-dimensional Hilbert spaces.

 

As I mentioned, "I don't even know how to ask this", I understand the concept of limits, but I've learned it in another language.

 

I mentioned "I'm not a maths student" to clarify that I'm not deeply interested in the intricacies of the very concept of limit, but more into a more pragmatic approach - as an engineer, one might say. Notice I asked for references, not to solve a particular problem. This page: http://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx

contains a series of limit identities, but nothing like I showed. Another example that consider is

[latex]\lim_{x\rightarrow a} f(x) \stackrel{?}{=} \lim_{x\rightarrow \frac{a}{b}} f(bx)[/latex]

is that always true, regardless of [latex]f(x)[/latex]?

[studiot]

 

If anything, Hilbert spaces are "well-behaved" - that's why I asked targeting those spaces instead of something more complicated. But if you want to simplify, we could go with finite-dimensional Hilbert spaces.

 

As I mentioned, "I don't even know how to ask this", I understand the concept of limits, but I've learned it in another language.

 

I mentioned "I'm not a maths student" to clarify that I'm not deeply interested in the intricacies of the very concept of limit, but more into a more pragmatic approach - as an engineer, one might say. Notice I asked for references, not to solve a particular problem. This page: http://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx

contains a series of limit identities, but nothing like I showed. Another example that consider is

[latex]\lim_{x\rightarrow a} f(x) \stackrel{?}{=} \lim_{x\rightarrow \frac{a}{b}} f(bx)[/latex]

is that always true, regardless of [latex]f(x)[/latex]?

 

OK, I'm glad you understand limits and that you have found Paul's Maths Pages.

 

Paul's Pages are to be recommended.

 

The page you link to is all about manipulation of limits as algebraic entities and a pretty comprehensive list of possible aglebraic operations on one, two or more limits and some constants.

 

You should note something he says at the top of his excellent table.

 

Assume that

and exist and that c is any constant.

 

 

The important bit is 'assume the limits exist'.

 

So here he sets out the conditions when the statements are true.

 

This is very important and often forgotten in applications.

It is up to the user to make sure that the usage is suitable for the application.

 

So yes, they are true, subject to that restriction.

 

 

Two important things you should remember from your understanding of limits.

 

Firstly the fact that a limit of a fucntion at x=a exists does not mean that the function has this value at x=a.

 

Secondly discontinuous functions may have different limits 'from the left' and 'from the right', depending upon which direction you approach the set point.

[Sriman Dutta]

For a general function [math]f(x):R \rightarrow R[/math] , and if as x tends to a, there exists a discrete value l which is approached by f(x), then by definition we have: [math]\lim_{x \rightarrow a} f(x) = l[/math]. For all x in the open interval [math](a+ \delta, a- \delta) [/math], the value [math] |f(x)-l|< \epsilon [/math], for all [math]\epsilon[/math] infinitely small.

If we want to make the 'tends to' part of limit as 0, then we consider h=x-a, and so, as [math]x \rightarrow a[/math], [math]h \rightarrow 0[/math].

This is a known technique of limit evaluation.

Edited May 27, 2017 by Sriman Dutta

[KipIngram]

 

I'm asking it here because I don't even know HOW to ask this!

 

Basically, I'm looking for some identities regarding the "tends to" part of the limit. So for instance:

[latex]\lim_{x\rightarrow b} f(x) = \lim_{x\rightarrow 0} f(x+b)[/latex]

Is the above expression correct? If so, is it always correct, at least for Hilbert spaces? What else can you do with the "tends to" part, can you rename variables there? References are welcome.

 

 

That looks a lot like a "change of reference frame." I.e., the left and right sides of the expression arise from one another by "relabeling" the numbers, but you didn't really change the function f since you "undo" the label change via its argument.

In both cases f is a function on the set of real numbers, so shifting your "reference" doesn't change anything fundamental.

[studiot]

For a general function [math]f(x):R \rightarrow R[/math] , and if as x tends to a, there exists a discrete value l which is approached by f(x), then by definition we have: [math]\lim_{x \rightarrow a} f(x) = l[/math]. For all x in the open interval [math](a+ \delta, a- \delta) [/math], the value [math] |f(x)-l|< \epsilon [/math], for all [math]\epsilon[/math] infinitely small.

If we want to make the 'tends to' part of limit as 0, then we consider h=x-a, and so, as [math]x \rightarrow a[/math], [math]h \rightarrow 0[/math].

This is a known technique of limit evaluation.

 

How about these functions?

 

[math]\left\{ \begin{array}{l}f(x) = 0,x \ne 0 \\f(x) = 1,x = 0 \\\end{array} \right\}[/math]

 

 

[math]\left\{ \begin{array}{l}f(x) = 0,x < 0 \\f(x) = 1,x = 0 \\f(x) = 2,x > 0 \\ \end{array} \right\}[/math]

 

[Sriman Dutta]

Hello studiot, the functions that you gave can't be written in the algebraic forms. In other words, these functions give their outputs, but doesn't show how the process gives the result. So I suspect we can't try that here.

[studiot]

Hello studiot, the functions that you gave can't be written in the algebraic forms. In other words, these functions give their outputs, but doesn't show how the process gives the result. So I suspect we can't try that here.

 

These were to illustrate point in post 8.

 

Why can't you apply your epsilon-delta definition to them, apart from the fact that your statement is a bit loose, since you should note that both epsilon and delta have to be strictly greater than zero. Also 'an infinitely small epsilon' is an arguably poor way to say 'for all epsilon greater than zero'.

 

A correct version of the definition is

 

The function f(x) approcaches a limit L as x approaches and stated

 

[math]\mathop {\lim }\limits_{x \to a} f(x) = L[/math]

 

 

If and only if

 

for every [math]\varepsilon > 0[/math] there is some [math]\delta > 0[/math] such that,

 

for all x, if

 

[math]0 < \left| {x - a} \right| < \delta [/math]

 

 

then

 

[math]\left| {f(x) - L} \right| < \varepsilon [/math]

 

Why does this not apply to show that the limit as x tends to zero of my first function is zero?

 

Can you see what the second function is designed to show and why I gave the value of f(0) = 1?

 

 

 

Edited May 28, 2017 by studiot

[Country Boy]

 

I'm asking it here because I don't even know HOW to ask this!

 

Basically, I'm looking for some identities regarding the "tends to" part of the limit. So for instance:

[latex]\lim_{x\rightarrow b} f(x) = \lim_{x\rightarrow 0} f(x+b)[/latex]

Is the above expression correct? If so, is it always correct, at least for Hilbert spaces? What else can you do with the "tends to" part, can you rename variables there? References are welcome.

 

 

Given that [latex]\lim_{x\to f} f(x)= L[/latex], let u= x- b. As x goes to b, u goes to 0. Since u= x- b then x= u+ b. So [latex]\lim_{u\to 0} f(u+ b)= L[/latex].

Edited May 30, 2017 by Country Boy

[inverse]

 

I'm asking it here because I don't even know HOW to ask this!

 

Basically, I'm looking for some identities regarding the "tends to" part of the limit. So for instance:

[latex]\lim_{x\rightarrow b} f(x) = \lim_{x\rightarrow 0} f(x+b)[/latex]

Is the above expression correct? If so, is it always correct, at least for Hilbert spaces? What else can you do with the "tends to" part, can you rename variables there? References are welcome.

 

 

this is not cogent when x=(x₁,x₂,x₃,...,x_n)