Understanding Epsilon-Delta Limits: A Guide for Solving Proofs

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In summary, Epsilon-Delta definition is a way to prove a limit by using a delta function. Epsilon and delta are both positive numbers, so the limit is always positive. However, if x is close to 4, the delta function will be smaller than 1, and the limit will be negative.
  • #1
Pete.Co.Lust
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Homework Statement


I want to ask a question about Epsilon-Delta definition...
I have already read a tons of definitions about Epsilon-Delta limit proof. But i am still stucking in some places...
E.g.) Prove it by using Espilon-Delta method, lim (x->4) x^3=64



Homework Equations



|x-4|<delta delta>0
|x^3|<espilon espilon>0

The Attempt at a Solution



I am kinda know what I should do, I changed |x^3-64| into |(x-4)|(x^2+4x+16)| and my objective afterward is to ensure| (x^2+4x+16)| is "small".
But i just don't know what I should do afterwards... Should I just Sub x=4 and then find out |(x^2+4x+16)|<48?

Plx help me :( ThX! :)
 
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  • #2
The point is not in showing that | (x^2+4x+16)| is "small, but that it is "bounded" when x is close to 4. So, restrict yourself to the region, for instance, 3<x<5. What can be said about | (x^2+4x+16)| in this region? Is it always smaller than some fixed number?
 
  • #4
arkajad,
Alright I got it. So u mean is to find out the max. number when x is "close" to 4? Afterwards we can state that |x-4|<1 (Because x is "close" to 4) and we can find the max. number of x is 5. we subsitute 5 into the equation and we have (5)^2+(4)(5)+16=61. And it would be espilon/61 afterwards?

If its espilon/61, we write this "delta=min{1,e/61}"? What does "delta=min{1,e/61}" actually means? ThankS

madah12,
The link is extremely useful for me to solve this problem. I found out the "solution"
(Which i don't know its right or not...) right there. Many thanks.
 
  • #5
min{1,e/61} means eiher 1 or e/61, whichever happens to be smaller.
 
  • #6
arkajad said:
min{1,e/61} means eiher 1 or e/61, whichever happens to be smaller.

So what do u mean is that if i pick e=62, the output will be 1 and if i pick e=60, the output will be 60/61? min{1,e/61}
 
  • #7
Yes, that's it. But are interested in "however small is epsilon" part, and for epsilon small enough (e<61) you will never use 1 as an output.
 
  • #8
Ohhhh you I got it! Thx for the explanation! It helped me muCH!
 

1. What is an Epsilon-Delta limit?

An Epsilon-Delta limit is a mathematical concept used in calculus to define the behavior of a function as it approaches a specific point. It involves using two variables, epsilon and delta, to represent the distance between the input and the desired point and the corresponding output and the desired limit, respectively.

2. How do you calculate an Epsilon-Delta limit?

To calculate an Epsilon-Delta limit, you must first choose a value for epsilon, typically a small positive number. Then, you must find a corresponding value for delta that will ensure that the distance between the input and the desired point is less than epsilon. Finally, you can use this delta value to show that the distance between the output and the desired limit is also less than epsilon.

3. Why are Epsilon-Delta limits important?

Epsilon-Delta limits are important because they provide a rigorous and precise way to define the behavior of a function at a specific point. They are crucial in the study of calculus and are used to prove theorems and solve problems related to continuity, differentiability, and convergence of sequences and series.

4. What are some common challenges when working with Epsilon-Delta limits?

One of the main challenges when working with Epsilon-Delta limits is choosing appropriate values for epsilon and delta. It can be difficult to determine the correct values, especially in more complex functions. Additionally, understanding the concept and its applications can also be challenging for some students.

5. How can Epsilon-Delta limits be applied in real-life situations?

Epsilon-Delta limits have many real-life applications, particularly in the fields of physics and engineering. For example, they are used in the study of motion, where delta represents time and epsilon represents error. They can also be used to model and analyze real-world phenomena, such as population growth or weather patterns.

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