Understanding formal definition of limits

In summary, the conversation discusses the formal definition of limits and the use of delta and epsilon notation in defining limits. The purpose of the formal definition is to describe a limit to someone who does not understand the meaning of the word. The correct definition of a limit states that for each epsilon greater than zero, there is a delta greater than zero such that the function falls within the interval (L-epsilon, L+epsilon) when x is in the interval (a-delta, a+delta). The conversation also poses a question about the possibility of having two epsilons and two deltas in the limit definition, which is proven to be equivalent to the correct definition.
  • #1
So far I could not understand the formal definition of limits. I know that limit of a function exists at a point say 'a' if left and right hand limit exist at that point. Then what is the need for the formal definition of limits?
Secondly, I am very confused by the usage of the notation of delta and epsilon and their usage in that definition.
 
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  • #2
that is defining something in terms of itself, considered circular. defining intelligence as the quality possessed by someone who is intelligent.

to define a limit you may not use the word limit in the definition. the point is to describe it to someone who does not understand the meaning of the word limit.

you have acomplished something, namely you have defiend a 2 sided limit for someone who already knows what a one sided limit is. now define a one sided limit and you will be done.
 
  • #3
oops actually your definition is wrong,a s the 2 one sided limits must exist AND be equal.
 
  • #5
I didn't want to start a new thread, but I have a question of my own:

I think essentially what the limit definition says is that
[tex]
\lim_{x\rightarrow a} f(x) = L
[/tex]

means for each [itex]\epsilon>0[/itex] there is a [itex]\delta>0[/itex] such that f(x) is in the inverval [itex](L-\epsilon, L+\epsilon)[/itex] whenever x is in the the interval [itex](a-\delta,a+\delta)[/itex] (x does not equal a).

Now, what I am wondering is: do the upper and lower bounds of the interval have to be the same distance from L and a?

Or could there be two epsilons and two deltas, making the limit definition:

[tex]
\lim_{x\rightarrow a} f(x) = L
[/tex]

means for each [itex]\epsilon_1>0[/itex] and [itex]\epsilon_2>0[/itex], there are [itex]\delta_1>0[/itex] and [itex]\delta_2>0[/itex] such that f(x) is in the inverval [itex](L-\epsilon_1, L+\epsilon_2)[/itex] whenever x is in the the interval [itex](a-\delta_1,a+\delta_2)[/itex] (x does not equal a).

Could this definition also work?
 
Last edited:
  • #6
The second 'definition' wouldn't make sense, unless I'm missing something, since the first states that for each [itex]\epsilon>0[/itex] ... So there must be more than one [itex]\epsilon>0[/itex].
 
  • #7
Hubert said:
I didn't want to start a new thread, but I have a question of my own:

I think essentially what the limit definition says is that
[tex]
\lim_{x\rightarrow a} f(x) = L
[/tex]

means for each [itex]\epsilon>0[/itex] there is a [itex]\delta>0[/itex] such that f(x) is in the inverval [itex](L-\epsilon, L+\epsilon)[/itex] whenever x is in the the interval [itex](a-\delta,a+\delta)[/itex] (x does not equal a).

Now, what I am wondering is: do the upper and lower bounds of the interval have to be the same distance from L and a?

Or could there be two epsilons and two deltas, making the limit definition:

[tex]
\lim_{x\rightarrow a} f(x) = L
[/tex]

means for each [itex]\epsilon_1>0[/itex] and [itex]\epsilon_2>0[/itex], there are [itex]\delta_1>0[/itex] and [itex]\delta_2>0[/itex] such that f(x) is in the inverval [itex](L-\epsilon_1, L+\epsilon_2)[/itex] whenever x is in the the interval [itex](a-\delta_1,a+\delta_2)[/itex] (x does not equal a).

Could this definition also work?

exercise - show these are equivalent, hint: I can take the minimum of two positive real numbers and get a positive real number.
 

1. What is a formal definition of limits?

A formal definition of limits is a mathematical concept used to describe the behavior of a function as its input approaches a certain value. It is a precise way to determine the value that a function approaches, or "limits" to, as its input gets closer and closer to a specific value.

2. How is a limit defined algebraically?

Algebraically, a limit is defined as the value that a function approaches as its input approaches a certain value, denoted as x. This is represented as the limit of f(x) as x approaches a, or limx→a f(x). It can be found by evaluating the function at values of x that are closer and closer to the value a.

3. What is the purpose of understanding formal definition of limits?

The purpose of understanding formal definition of limits is to have a precise and rigorous way to describe the behavior of a function at certain points. It allows for the evaluation of limits at any point, even when the function may not be defined at that point. This concept is crucial in calculus and is used to solve various problems in mathematics and science.

4. What are the three components of a formal definition of limits?

The three components of a formal definition of limits are the limit value, the input value, and the difference between the two. The limit value is the value that the function approaches, the input value is the point at which the limit is being evaluated, and the difference is the distance between the two values.

5. How is the concept of limits used in real-world applications?

The concept of limits is used in many real-world applications, such as calculating the velocity of an object, determining the maximum and minimum values of a function, and finding the area under a curve. It is also used in fields such as economics, physics, and engineering to model and analyze various systems and phenomena.

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