Understanding the basic concept of a Limit

[nycmathguy]

TL;DR Summary

Basic concept of a limit.

Hello everyone. How are you? I want to learn calculus so badly. I plan to do a self-study of calculus l, ll, and lll. Before thinking so far ahead, I need a clear, basic definition of the concept of a limit. Textbook language is never easy to grasp unless the student is gifted. I am not gifted mathematically but love the subject.

I understand the limit idea to be the following:

•The value a function is tending to without actually getting there.

•The height a function is trying to reach in terms of the y-axis.

•In terms of a point (x, y), the limit is (value the function is tending to, limit). In other words, x = the value a function is tending to and y = the limit, the height of the function.
Is my understanding of a limit clear or not?

 

[fresh_42]

You are sounding a bit confused. E.g. "without getting there" is wrong: $\lim_{x \to 2} x^2=4$ so the function $f(x) =x^2$ is actually getting to the limit $4.$ Things are getting a bit more complicated if infinities, or holes come into play.

$L:=\lim _{x\to a} f(x)$ describes the value of $f(x)$ if $x$ approaches $a$ on some path. It does not say whether $f(a)$ is defined or not. All cases can occur: $L=\infty \, , \,L=-\infty \, , \,L=f(a) \in \mathbb{R}$ or $f(a)$ is undefined, although the limit $L$ exists (a gap in the graph of $f$ at position $x=a$).

 

[Delta2]

I think you might confusing two different things:

1. The value that the independent variable $x$ is tending to, which we may refer to as $x_0$
2. The value that the dependent variable $y=f(x)$ is tending to (this is the value the function is tending to and it is equal to the limit L)

So when we write $$\lim_{x\to 2}x^2=4$$ we mean that , AS the value of the independent variable $x$ is tending to 2 ($x_0=2$ in this case), the value of the dependent variable $y=x^2$ is tending to the limit L=4.

 

[nycmathguy]

I think you might confusing two different things:

1. The value that the independent variable $x$ is tending to, which we may refer to as $x_0$
2. The value that the dependent variable $y=f(x)$ is tending to (this is the value the function is tending to and it is equal to the limit L)

So when we write $$\lim_{x\to 2}x^2=4$$ we mean that , AS the value of the independent variable $x$ is tending to 2 ($x_0=2$ in this case), the value of the dependent variable $y=x^2$ is tending to the limit L=4.

As x tends a number, y = f(x) is approaching the limit L.

 

[Delta2]

As x tends a number, y = f(x) is approaching the limit L.

Yes this is absolutely correct.

 

[phinds]

In terms of a point (x, y), the limit is (value the function is tending to, limit). In other words, x = the value a function is tending to and y = the limit, the height of the function.

You want to be careful about limiting your thinking (no pun intended) to a particular axis orientation. A limit can be on either axis depending on how a function is defined and for that matter limits are not confined to 2D.

 

[nycmathguy]

You are sounding a bit confused. E.g. "without getting there" is wrong: $\lim_{x \to 2} x^2=4$ so the function $f(x) =x^2$ is actually getting to the limit $4.$ Things are getting a bit more complicated if infinities, or holes come into play.

$L:=\lim _{x\to a} f(x)$ describes the value of $f(x)$ if $x$ approaches $a$ on some path. It does not say whether $f(a)$ is defined or not. All cases can occur: $L=\infty \, , \,L=-\infty \, , \,L=f(a) \in \mathbb{R}$ or $f(a)$ is undefined, although the limit $L$ exists (a gap in the graph of $f$ at position $x=a$).

This idea is totally new to me. I am going through the textbook on my own. No specific reason other than learning the material. Let's stick to the basics of calculus by avoiding math jargon and complicated symbols that mean nothing to a novice calculus self-study individual.

Here is a better definition:

As x tends a number, y = f(x) is approaching the limit L.

You say?

 

[nycmathguy]

You want to be careful about limiting your thinking (no pun intended) to a particular axis orientation. A limit can be on either axis depending on how a function is defined and for that matter limits are not confined to 2D.

This idea is totally new to me. I am going through the textbook on my own. No specific reason other than learning the material. Let's stick to the basics of calculus by avoiding math jargon and complicated symbols that mean nothing to a novice calculus self-study individual.

Here is a better definition:

As x tends a number, y = f(x) is approaching the limit L.

You say?

 

[Delta2]

You want to be careful about limiting your thinking (no pun intended) to a particular axis orientation. A limit can be on either axis depending on how a function is defined and for that matter limits are not confined to 2D.

He might me limiting his thinking, but you expanding it too much, by start talking about multivariable calculus when the OP said he just starting learning calculus I.

 

[PeroK]

Here is a better definition:

As x tends a number, y = f(x) is approaching the limit L.

You say?

I say that doesn't even make sense. Here's an informal definition:

https://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx

And, a formal one:

https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx

 

[Delta2]

I say that doesn't even make sense

this is some sort of overstatement by you. I read the first link and I find that what the OP says at #8 is very similar to what the link says. Why do you say it doesn't even make sense?

 

[PeroK]

this is some sort of overstatement by you. I read the first link and I find that what the OP says at #8 is very similar to what the link says. Why do you say it doesn't even make sense?

Because it should be:

If as $x$ tends to $a$, $f(x)$ tends to $L$, then we say that $L$ is the limit of the function $f(x)$ as $x$ tends to $a$; and, we write: $$\lim_{x \rightarrow a}f(x) = L$$ Trying to improve on this is a mathematical dead-end, IMO.

 

[PeroK]

I understand the limit idea to be the following:

•The value a function is tending to without actually getting there.

There's nothing in the definition of a limit that says that it doesn't "actually get there".

First, if we have a constant function $f(x) = c$, then we have (for any $a$) $$\lim_{x \rightarrow a} f(x) = c$$ and we see that the limit is attained.

Also, we have: $$\lim_{x \rightarrow 0} x \sin \frac 1 x = 0$$ and the function $f(x) = x \sin \frac 1 x$ attains the limit value $0$ an infinite numbers of times for $x \ne 0$.

 

[Delta2]

Because it should be:

If as $x$ tends to $a$, $f(x)$ tends to $L$, then we say that $L$ is the limit of the function $f(x)$ as $x$ tends to $a$; and, we write: $$\lim_{x \rightarrow a}f(x) = L$$ Trying to improve on this is a mathematical dead-end, IMO.

I must be brain dead cause I also find what you saying here very similar to what the OP says at #8, you just expand it a bit further.

 

[PeroK]

I must be brain dead cause I also find what you saying here very similar to what the OP says at #8, you just expand it a bit further.

Have a cup of coffee and read them both again!

 

[nycmathguy]

There's nothing in the definition of a limit that says that it doesn't "actually get there".

First, if we have a constant function $f(x) = c$, then we have (for any $a$) $$\lim_{x \rightarrow a} f(x) = c$$ and we see that the limit is attained.

Also, we have: $$\lim_{x \rightarrow 0} x \sin \frac 1 x = 0$$ and the function $f(x) = x \sin \frac 1 x$ attains the limit value $0$ an infinite numbers of times for $x \ne 0$.

I thank you for your reply. However, I am looking at calculus for the first time outside of a classroom setting. I am slowly going through the textbook one chapter, one section at a time. The limit of trig functions is still a few chapters and sections ahead.

 

[fresh_42]

I thank you for your reply. However, I am looking at calculus for the first time outside of a classroom setting. I am slowly going through the textbook one chapter, one section at a time. The limit of trig functions is still a few chapters and sections ahead.

Draw some graphs of possible functions and look at what happens to the graph if $x$ approaches certain points. E.g.

 

[jedishrfu]

For more details on limits checkout the 3blue1brown youtbe sequence called the Essence of Calculus. Its ten short 10min videos covering this and other key concepts.

 

[nycmathguy]

Draw some graphs of possible functions and look at what happens to the graph if $x$ approaches certain points. E.g.

View attachment 284372

We will get to graphs as I journey through the textbook. I'm excited about calculus events my PSA level is 4.9. Pray for me.

 

[berkeman]

my PSA level is 4.9

That's a bit high for your age. Are you under a doctor's care for it?

 

[nycmathguy]

That's a bit high for your age. Are you under a doctor's care for it?

Yes, my primary care doctor at the VA Hospital wants me back in 6 weeks for another PSA. There are men with higher PSA levels that do not have cancer. I am not going to quickly run into a prostate biopsy situation. Let's see what happens in 6 weeks. I'll keep you posted. Thank you so much for asking.

 

[romsofia]

This video, and channel overall, should help you on your journey (patrickJMT has some of the BEST videos on computational calculus IMO):

 

[nycmathguy]

This video, and channel overall, should help you on your journey (patrickJMT has some of the BEST videos on computational calculus IMO):

Thanks for the video clip. I will check it out.

 

[mathwonk]

I think I understand where you are coming from with the "without getting there" language because I used to actualky use that in teaching limits, although I sympathize with opponents of it.

Take an example of a limit if a sequence, (which is a function defined on the positive integers), like f(n) = 1/n. We ask for "the limit of f(n) as n goes to infinity". To me, the entries in the sequence are approximating values, and the limit is the number they are trying to approximate. So ask yourself what number do the numbers 1/n get closer to as n gets larger and larger. The answer is zero. That is, if you go out far enough in the sequence, the elements get, and remain, as close to zero as you might wish. I.e. if you want them to be closer than .0001, then after n gets larger than 10,000, every later sequence entry 1/n will be closer to zero than .0001. So zero is the limit of the sequence. In this case no element of the sequence ever equals zero, so although no entry in the sequence actually gets to zero, every element, far enough out, gets as close as desired.

Now here is another sequence: f(1) =1, f(2) = 2, f(3) = 3, f(4) = 4, f(5)= 1/5, f(6) = 1/6, f(7) =0, f(8) = 0, f(9) = 0, ...f(n) = 0 for every n ≥ 9. This sequence fools around for a while getting larger, but soon turns around and starts getting smaller, and all of a sudden jumps to zero, where it remains forever after. Since the only thing that matters in a sequence is where it eventually goes, this sequence is also (eventually) approximating to, i.e. heding ultimtely toward, zero. Indeed if you wnt to get an approximation to within .000001, of zero, you can take any sequence entry further out than f(7). So here again the limit is zero. But in this case, most elements of the sequence do actually get to zero, the limit.

So sometimes the sequence actualy gets to the limit "early", and sometimes it does not. This doies not matter at all for the notion of a limit. The limit is the number that the sequence ultimately zeroes in on, whether it actually gets there earlier or not does not matter.

I have to go now, but the case of functions is similar. I.e. when looking at the limit of a function like f(x) = (x^2-4)/(x-2), as x goes to 2, we are interested in what number is being approximated by these values f(x) when x is very near 2, but not equal to 2. That is because the function is not defined at 2. This is analogous to the fact the sequence function above was not defined at n = infinity.

It is more confusing to consider functions like f(x) = x^2, and ask for the ,imit as goes to 2, because this function is defined at 2, i.e. f(2) = 4. Since the limit of this function as x goes to 2, is also 4, you might say that the function does get to its limit, namely the limit as x goes to 2, is whatever number x^2 is fgettoing closer to as x gets closer to 2, and this is 4. But you are supposed to compute the limit only by looking at numbers x with x ≠ 2, so really, in my opinion this function also does not "get to 4", at least not as long as x has not got to 2.

The link is that some functions are defined at the point that x is going to, i.e. this f(x) = x^2 is defined at x=2, and one can ask two separate questions: 1) is there a limit (and what is it), of x^2, as x goes to 2 without reaching it? 2) does that limit, equal the value of f(2)? If so, the function is a good function, called continuous.

got to go.

 

[nycmathguy]

I think I understand where you are coming from with the "without getting there" language because I used to actualky use that in teaching limits, although I sympathize with opponents of it.

Take an example of a limit if a sequence, (which is a function defined on the positive integers), like f(n) = 1/n. We ask for "the limit of f(n) as n goes to infinity". To me, the entries in the sequence are approximating values, and the limit is the number they are trying to approximate. So ask yourself what number do the numbers 1/n get closer to as n gets larger and larger. The answer is zero. That is, if you go out far enough in the sequence, the elements get, and remain, as close to zero as you might wish. I.e. if you want them to be closer than .0001, then after n gets larger than 10,000, every later sequence entry 1/n will be closer to zero than .0001. So zero is the limit of the sequence. In this case no element of the sequence ever equals zero, so although no entry in the sequence actually gets to zero, every element, far enough out, gets as close as desired.

Now here is another sequence: f(1) =1, f(2) = 2, f(3) = 3, f(4) = 4, f(5)= 1/5, f(6) = 1/6, f(7) =0, f(8) = 0, f(9) = 0, ...f(n) = 0 for every n ≥ 9. This sequence fools around for a while getting larger, but soon turns around and starts getting smaller, and all of a sudden jumps to zero, where it remains forever after. Since the only thing that matters in a sequence is where it eventually goes, this sequence is also (eventually) approximating to, i.e. heding ultimtely toward, zero. Indeed if you wnt to get an approximation to within .000001, of zero, you can take any sequence entry further out than f(7). So here again the limit is zero. But in this case, most elements of the sequence do actually get to zero, the limit.

So sometimes the sequence actualy gets to the limit "early", and sometimes it does not. This doies not matter at all for the notion of a limit. The limit is the number that the sequence ultimately zeroes in on, whether it actually gets there earlier or not does not matter.

I have to go now, but the case of functions is similar. I.e. when looking at the limit of a function like f(x) = (x^2-4)/(x-2), as x goes to 2, we are interested in what number is being approximated by these values f(x) when x is very near 2, but not equal to 2. That is because the function is not defined at 2. This is analogous to the fact the sequence function above was not defined at n = infinity.

It is more confusing to consider functions like f(x) = x^2, and ask for the ,imit as goes to 2, because this function is defined at 2, i.e. f(2) = 4. Since the limit of this function as x goes to 2, is also 4, you might say that the function does get to its limit, namely the limit as x goes to 2, is whatever number x^2 is fgettoing closer to as x gets closer to 2, and this is 4. But you are supposed to compute the limit only by looking at numbers x with x ≠ 2, so really, in my opinion this function also does not "get to 4", at least not as long as x has not got to 2.

The link is that some functions are defined at the point that x is going to, i.e. this f(x) = x^2 is defined at x=2, and one can ask two separate questions: 1) is there a limit (and what is it), of x^2, as x goes to 2 without reaching it? 2) does that limit, equal the value of f(2)? If so, the function is a good function, called continuous.

got to go.

Interesting. Thank you.

 

[Stephen Tashi]

Summary:: I need a clear, basic definition of the concept of a limit. Textbook language is never easy to grasp unless the student is gifted.

We have to distinguish between a clear intuitive concept of a limit versus the formal definition of a limit, which is perhaps what you encounter in your textbook. The formal definition is the only definition that is reliable for purpose of proving things about limits. The intuitive definition is what most students learn first.

Furthermore, the phrase "definition of limit" is ambiguous. There are different types of limits. The outstanding feature of the intutiive definitions of limits is the concept of a process taking place in time or in steps. In the formal definition of limit there is no mention or need of such a process. For example, the formal definition (or its notation) may contain language such as "as x approaches", however there is nothing in the formal definition that defines a process called "approaching". If you need to work problems in a textbook that require you to understand the formal definition (the so-called "epsilon-delta definition") of a limit, then you have to progress beyond the intuitive definition, but the intuitive definition is where you should start.

 

[jedishrfu]

Since the op has left the site, it’s a good time to close this thread.

Thank you all for contributing here.