Precise definition of the limit (epsilon-delta)

In summary: It is decreasing for...what values of x?Given an epsilon, what is the largest value that ##|f(x) - 18| < \epsilon## can take? How big must delta be to ensure that |f(x) - 18| < epsilon when |x - 2| < delta? In summary, The conversation is discussing the definition of limit and a specific question on rearranging the expression |(4x^2+2)-18| to look like |x-2|. The conversation includes discussions on using an upper bound for x+2, using the triangle inequality, and solving for delta in terms of epsilon. The goal is to find a delta value that satisfies the
  • #1
Cal124
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Sorry, I am really struggling with the precise definition of the limit. I have a specific question I'm trying to work out
lim(x->2) (4x2+2)=18

skipping the introduction part
any advice? I am just not sure how to get rid of the 2 value to re-arrange |(4x2+2)-18| to look like |x-2|

|x-2|<delta |(4x2+2)-18|<epsilon
|(4x2-16|<epsilon

I did go down the route of
|(4x2+2)-18| = |2x+4| |2x-4|<epsilon
but this step just lead me to the same problem
 
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  • #2
So you realize that you have [itex]|2x- 4||2x+ 4|< \epsilon[/itex]. You should be immediately able to see that this is the same as [itex]4|x- 2||x+ 2|< \epsilon[/itex] and then [itex]|x-2||x+2|< \epsilon/4[/itex].

And you want "[itex]|x- 2|< \delta[/itex]". Now, if that "|x+ 2|" were a positive constant, you could just divide by it: [itex]|x- 2|< \epsilon/(4|x+2|)[/itex] but because it contains "x" you can't do that. What you can do is replace |x+2| with an constant upper bound. If we could say that |x+ 2|< U, for x close to 2, then 1/U< 1/|x+2| and [itex]\epsilon/(4U)< \epsilon/4|x+ 2|)[/itex].

If x is close to 2, we can certainly say that -1< x- 2< 1 (choosing -1 and 1 just because they are easy) so that 4- 1= 3< x+ 2> 4+ 1= 5. An upper bound on x+ 2 is 5.
 
  • #3
Not sure if this is right but thought I'd post my progress along with helpful sites in case somebody else has the same problem
http://math.stackexchange.com/quest...-where-i-let-delta-be-a-minimum-of-two-values
http://mathforum.org/library/drmath/view/53738.html
(not sure on the forums policy on links, so apologies if this isn't right)

lim(x->2) 4x^2+2) = 18

0<|x-2|<delta

|(4x^2+2)-18|<epsilon
|(4x^2-16|<epsilon
|2x+4|.|2x-4|<epsilon
/2
|x+2|.|x-2|<epsilon/2
1<|x+2|<3
-4
-3<|x-2|<-1
|x-2|<-1

|x+2|.|x-2|<delta.-1
hence
epsilon = (-)delta ~ (-)epsilon = delta
 
  • #4
Your claim is that given an epsilon>0, then there is a delta>0 such that if |x-2|< delta, |(4x^2+2) - 18| < epsilon.
You have |x-2| < delta and above you wrote |x+2| |x-2| < epsilon/2.
HInt: ##|x-2 + 4 | \leq | x-2| + 4##.
Solve for delta in terms of epsilon.
Also...I disagree with where you wrote 1< |x+2|< 3 since the assumption is that x is getting close to 2.
 
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  • #5
RUber said:
Your claim is that given an epsilon>0, then there is a delta>0 such that if |x-2|< delta, |(4x^2+2) - 18| < epsilon.
You have |x-2| < delta and above you wrote |x+2| |x-2| < epsilon/2.
HInt: ##|x-2 + 4 | \leq | x-2| + 4##.
Solve for delta in terms of epsilon.
Also...I disagree with where you wrote 1< |x+2|< 3 since the assumption is that x is getting close to 2.
That's the part I don't truly understand to be honest. The guide I read suggested as your trying to stay witching a small range 1 either side of the value (2) |x+2| is this wrong or have I (the more likely) misunderstood
Thanks but I don't quite get your hint
 
  • #6
Cal124 said:
That's the part I don't truly understand to be honest. The guide I read suggested as your trying to stay witching a small range 1 either side of the value (2) |x+2| is this wrong or have I (the more likely) misunderstood
Thanks but I don't quite get your hint
They're probably trying to use the fact that if ##\delta<1## and ##|x-2|<\delta##, then |x+2| is less than...what? You can probably figure that out for yourself. This simplifies the problem. The idea here is that we don't need to find the largest ##\delta## that works, so we might as well use that freedom to choose a ##\delta## that's small enough to simplify the calculations.

RUber is using the rewrite x+2=x-2+4 and the triangle inequality ##|x+y|\leq|x|+|y|## to find an inequality of the form ##|x+2|\leq f(\delta)## (where f is some function) that holds when ##|x-2|<\delta##. (It's up to you to figure out what f is). Then we can say that for all x such that ##|x-2|<\delta##, we have
$$|4x^2+2-18|=4|x+2||x-2|<\text{what?}$$ Since you need this to be less than ##\varepsilon##, you can find the largest ##\delta## that works by solving the equation ##\text{what?}=\varepsilon## for ##\delta##.
 
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  • #7
HallsofIvy said:
So you realize that you have [itex]|2x- 4||2x+ 4|< \epsilon[/itex]. You should be immediately able to see that this is the same as [itex]4|x- 2||x+ 2|< \epsilon[/itex] and then [itex]|x-2||x+2|< \epsilon/4[/itex].

Why is this 4? Is it not 2 or am i seeing this wrong?

Fredrik said:
They're probably trying to use the fact that if ##\delta<1## and ##|x-2|<\delta##, then |x+2| is less than...what?

less than 1?

Fredrik said:
RUber is using the rewrite x+2=x-2+4 and the triangle inequality ##|x+y|\leq|x|+|y|## to find an inequality of the form ##|x+2|\leq f(\delta)## (where f is some function) that holds when ##|x-2|<\delta##. (It's up to you to figure out what f is). Then we can say that for all x such that ##|x-2|<\delta##, we have
$$|4x^2+2-18|=4|x+2||x-2|<\text{what?}$$ Since you need this to be less than ##\varepsilon##, you can find the largest ##\delta## that works by solving the equation ##\text{what?}=\varepsilon## for ##\delta##.

I'm struggling with the, almost, canceling down from factoring to
[itex]|x-2||x+2|< \epsilon/2[/itex]
from this step I get really lost, as far as I'm aware the |x-2| is controlled by delta so i have to work with |x+2|
by this i use the fact i need to stay close to 2 so taking 1 either side i can say 1<|x+2|<3 (either side of the 2) and to make it into |x-2| i minus 4 giving -3<|x-2|<-1
which gives me -1 which is the upper? so as
|x-2| |x+2|<epsilon/2
|x-2| |x+2|<delta . epsilon/-2

now I just feel lost. I'm really not sure where I am going wrong
thanks for the help though
 
  • #8
Cal124 said:
i can say 1<|x+2|<3 (either side of the 2) and to make it into |x-2| i minus 4 giving -3<|x-2|<-1
which gives me -1 which is the upper?

First, look at what you've done. That's a very basic error in using the absolute value.

Perhaps this epsilon-delta thing looks like some sort of mathematical black magic; whereas, it is simply algebra and estimation. To try to get a grip on what is happening here, I suggest you calculate the function values for ##|x-2| < \delta## for some specific values of ##\delta##.

Note first that ##f(x) = 4x^2 + 2## is increasing for ##x > 0##. So, first we assume that ##\delta < 2## so that ##x## is always positive.

For ##\delta = 1##, we have ##|x-2| < 1 \ \Rightarrow \ 1 < x < 3 \ \Rightarrow \ f(1) < f(x) < f(3) \ \Rightarrow \ 6 < f(x) < 38 \ \Rightarrow \ |f(x) - 18| < 20##

Now try with ##\delta = 1/2## and ##\delta = 1/4##.

At least this might give you an idea of what's going on. The next step, of course, is to try to work out how small ##\delta## has to be to keep ##|f(x) - 18| < \epsilon##, where ##\epsilon## is an arbitrary number.

It strikes me, looking at your posts, that you haven't really understood how the algebraic steps with general epsilon and delta relate to the simple functional evaluations for various values of x.

In fact, you could probably gain some insight here by drawing a graph of the function as well, so you can see what's going on when you keep x "close" to 2.
 
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  • #9
Cal124 said:
Why is this 4? Is it not 2 or am i seeing this wrong?
I am not sure what you are seeing but you have |2x+ 4||2x- 4|= |2(x+ 2)||2(x- 2|= 2|x+ 2|(2)|x- 2|= 4|x+ 2||x- 2|.

less than 1?
I'm struggling with the, almost, canceling down from factoring to
[itex]|x-2||x+2|< \epsilon/2[/itex]
from this step I get really lost, as far as I'm aware the |x-2| is controlled by delta so i have to work with |x+2|
by this i use the fact i need to stay close to 2 so taking 1 either side i can say 1<|x+2|<3 (either side of the 2) and to make it into |x-2| i minus 4 giving -3<|x-2|<-1
which gives me -1 which is the upper? so as
You just said you "have to work with |x+ 2|" so why are you looking at |x- 2|? We want to be able to say that "if [itex]|x- 2|< \delta[/itex]- that is if x is close enough to 2 then [itex]|(x- 2)(x+ 2)|< \epsilon[/itex]. In particular we can choose to look at x between 1 and 3- so that x is at least that "close to 2". (x, NOT x+ 2, between 1 and 3) then since 1< x< 3, 3< x+ 2< 5.

|x-2| |x+2|<epsilon/2
|x-2| |x+2|<delta . epsilon/-2

now I just feel lost. I'm really not sure where I am going wrong
You appear to be doing things backwards! Surely you see that if 1< x+ 2< 3 then -1< x< 1 so x is not anywhere near the "target" of x going to 2?

thanks for the help though
 
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  • #10
Cal124 said:
less than 1?
You said that something about "...trying to stay witching a small range 1 either side of the value (2)". That sentence isn't easy to understand, but I interpreted it as keeping x at a distance less than 1 from the number 2, i.e. as requiring that |x-2|<1.

The number 1 is just a convenient choice that makes it easy to handle the |x+2| factor.

Cal124 said:
I'm struggling with the, almost, canceling down from factoring to
[itex]|x-2||x+2|< \epsilon/2[/itex]
The thing that you need to be less than ##\varepsilon## is
$$|4x^2+2-18|=|4x^2-16|=4|x^2-4|=4|x+2||x-2|,$$ so I think you're off by a factor of 2.

Cal124 said:
from this step I get really lost, as far as I'm aware the |x-2| is controlled by delta so i have to work with |x+2|
The fact that |x-2| is "controlled by delta" ensures that |x+2| is too. You can see this by rewriting |x+2| as |(x-2)+4| and using the triangle inequality. A more intuitive (but not as rigorous) way is to draw a line to represent the real numbers, and mark the numbers -2 and 2. |x-2| is the distance between x and 2. |x+2| is the distance between x and -2. If x is at a distance from 2 that's less than ##\delta##, then what is the least upper bound on the distance between x and -2?
 
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  • #11
This looks like a textbook example of what this part of the forum is not supposed to be used for, because all I see are questions on questions, all of the type "what do you mean by this, what do you mean by that?" Cal124's last post is exactly like this. That thing about 4 versus 2, I mean come on.

Cal124 must put the effort in. These limit problems are not hard but I don't see any serious thought being spent on it by him and for that reason, I think it is in the wrong place and the help has been plentiful but taken for granted, so far as I can tell.
 
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  • #12
verty said:
This looks like a textbook example of what this part of the forum is not supposed to be used for, because all I see are questions on questions, all of the type "what do you mean by this, what do you mean by that?" Cal124's last post is exactly like this. That thing about 4 versus 2, I mean come on.

Cal124 must put the effort in. These limit problems are not hard but I don't see any serious thought being spent on it by him and for that reason, I think it is in the wrong place and the help has been plentiful but taken for granted, so far as I can tell.

Sounds like maybe this forum isn't for me then, maybe when I'm better educated. Nothing was taken for granted, thanks to those who did help, I will study your advice and hopefully get it.
 
  • #13
Cal124 said:
Sounds like maybe this forum isn't for me then, maybe when I'm better educated. Nothing was taken for granted, thanks to those who did help, I will study your advice and hopefully get it.
We don't need you to be better educated. We just need you to do a bigger part of the work towards a solution. We're supposed to give you hints so that you can solve the problem for yourself, not solve the problem for you. But what happened here was that we gave you a piece of the solution, and then you asked questions until you got the next piece, and so on. This isn't just your mistake. Since you're new here, I'd say that we are more to blame for this than you.
 
  • #14
I apologize for jumping in so brashly. You can put it down to me having a bad day.
 

Related to Precise definition of the limit (epsilon-delta)

1. What is the precise definition of the limit?

The precise definition of the limit, also known as the epsilon-delta definition, is a mathematical concept used to describe the behavior of a function as the input approaches a certain value. It states that the limit of a function at a particular point is the value that the function approaches as the input gets closer and closer to that point.

2. Why is the epsilon-delta definition important?

The epsilon-delta definition is important because it provides a rigorous and exact way to define the concept of a limit. It allows us to make precise statements about the behavior of a function and provides a foundation for more advanced mathematical concepts such as continuity and differentiability.

3. How is the epsilon-delta definition used to prove the existence of a limit?

To prove the existence of a limit using the epsilon-delta definition, we must show that for any small positive value of epsilon, we can find a corresponding value of delta such that the distance between the input and the point at which we want to find the limit is less than delta, and the distance between the output and the limit is less than epsilon. This shows that the function approaches the limit as the input approaches the desired point.

4. Can the epsilon-delta definition be used for all functions?

Yes, the epsilon-delta definition can be used for all functions, as long as they are defined at the point in question. It is a general concept that applies to all types of functions, including polynomial, exponential, trigonometric, and more.

5. How is the epsilon-delta definition related to the concept of continuity?

The epsilon-delta definition is closely related to the concept of continuity. A function is said to be continuous at a point if its limit at that point exists and is equal to the function's value at that point. In other words, the epsilon-delta definition helps us determine if a function is continuous at a particular point by showing that it approaches the same value as the input gets closer to that point.

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