Help with formal definition of the limit of a function

In summary, the conversation discusses the concept of limits and how they are formally defined using the epsilon-delta method. This method involves finding a relationship between delta and epsilon to show that the limit of f(x) approaches a specific value as x approaches a certain number. It is not a circular argument, as it may seem, but instead a way to carefully analyze and prove the existence of a limit. However, in some cases, the limit may not exist, making this approach ineffective.
  • #1
j-lee00
95
0
The problem is not to conduct the proof but how the proof works. (It seems to be a circular argument)



Please use laymans english to explain, thank you
 
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  • #2
what are you asking?
 
  • #3
how the proof works using the formal definition of a limit
 
  • #4
with epsilon and delta?
 
  • #5
yep that's correct
 
  • #6
Formal Definition, so yup. Never really liked epsilon delta proofs myself, but anyway..

The proof works by showing that in the close neighborhood of the value that the limit is approaching, all the values in the neighborhood as becoming the same.
 
  • #7
ok, firstly, you ask yourself how may you otherwise define a "limit"? if you think about it you will see that the formal definition of a limit make perfect sense... it is indeed what we meant by a "limit" that is: we want to see what happen to f(x) as x tends to a certain number x_0.
suppose we want to show
[tex]\lim_{x\rightarrow x_0} f(x) = L[/tex],
then we would like to know whether f(x) indeed gets close to L as x gets close to x_0. And that is exactly what the definition is doing.

now in proofs, you often want to find a relationship between [tex]\delta[/tex] and [tex]\epsilon[/tex]; and often what that does is to simply ensure that we can ALWAYS find the appropriate values for them so that the the inequalities involving [tex]\delta[/tex] and [tex]\epsilon[/tex] can be satisfied.
 
  • #8
thank you very much
 
  • #9
When you say "It seems to be a circular argument" you may be thinking of this:

To prove that [itex]lim_{x\rightarrow 3} 2x+ 1= 7[/itex] a typical argument goes "If [itex]|f(x)-L|= |2x+1- 7|< \epsilon[/itex] then [itex]\|2x- 6|= 2|x-3|< \epsilon[/itex] so it suffices to take [itex]|x-3|< \delta= \epsilon/2[/itex]".

That seems "circular" because we start from [itex]|f(x)-L|< \epsilon[/itex] which, by the definition of limit, is what we want to show. You are correct that that is not a formal proof- it is more like analyzing the problem to deciding HOW to write a proof- It is deciding how we should choose [itex]\delta[/itex] in order to get the result we want. A "true" proof would go the other way:
"Given [itex]\epsilon[/itex], take [itex]\delta= \epsilon/2[/itex] (which looks like the professor is picking it out of the air since the student didn't see the analysis above). Then if [itex]0< |x-3|< \delta= \epsilon/2[/itex], we have [itex]0< 2|x-3|= |2x- 6|= |2x+ 1- 7|< \epsilon[/itex]".

Typically, having done the "analysis" we don't need to write out the "true" proof because it is clear that every step in going from [itex]\epsilon[/itex] to [itex]\delta[/itex] is reversible- it's obvious that we can go from [itex]\delta[/itex] to [itex]\delta[/itex]. A "proof" where you go from the conclusion to the hypothesis by reversible steps, so that it is obvious you could go form hypothesis to conclusion, is sometimes called "synthetic proof". It's used a lot, for example, in proving trig identities.
 
  • #10
the perception of a "circular argument" appearing only because in those cases (probably cases that you have been exposed to so far) it DOES work and that limit exists and does go to the number we want to proof. But i think all you need to see is an example of a limit that doesn't exist... and you will see that in this instance, you just can't find a delta and epsilon that work, no matter how hard you try... hence, this "circular argument" does not appear here. It looked like so before because it worked.
 

1. What is the definition of the limit of a function?

The formal definition of the limit of a function is the value that the function approaches as the input (x) gets closer and closer to a specific value (a). This is denoted by the notation lim f(x) = L, where L is the limit and x approaches a.

2. How is the limit of a function different from the value of the function at a specific point?

The limit of a function is the value that the function approaches as x gets closer to a specific value. It is not necessarily the same as the value of the function at that specific point. The limit is a theoretical concept, while the value at a specific point is the actual output of the function at that point.

3. What is the importance of understanding limits in calculus?

Limits are a fundamental concept in calculus and are essential for understanding the behavior of functions. They allow us to determine the rate of change of a function, the existence of derivatives, and the convergence or divergence of infinite series. Limits are also used in many real-world applications, such as optimization and optimization problems.

4. Can you explain the concept of a one-sided limit?

A one-sided limit is a limit where x only approaches the specified value from one side. This means that the value of the function as x approaches the specified value from the left or right may be different. One-sided limits are used when there is a discontinuity in the function or when the function is defined differently on either side of the specified value.

5. How do you calculate the limit of a function algebraically?

To calculate the limit of a function algebraically, you can use the limit laws, which state that the limit of a sum, difference, product, or quotient of two functions is equal to the sum, difference, product, or quotient of the limits of the individual functions. You can also use substitution, where you substitute the specified value into the function and simplify. However, in some cases, you may need to use more advanced techniques, such as L'Hopital's rule or the squeeze theorem, to evaluate the limit.

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