Proving with Delta Epsilon: A Beginner's Guide

In summary, the conversation discusses the difficulty in understanding how to prove limits using the delta-epsilon definition. The problem of proving a limit involving a sum of two functions without assuming the limit of the sum is equal to the sum of their individual limits is also mentioned. The suggestion of using the same proof for limit arithmetics and specifically applying it to the given problem is given. A hint to use ε/2 is also provided.
  • #1
conorsmom
1
0
Hi there, I'm having trouble understanding how to prove things using the [tex]\delta[/tex] [tex]\epsilon[/tex] definition. I have read a few other threads and sites, but I can't seem to put it together. For example, I came across this problem, if given limx-->af(x) = L, how would I prove (using delta-epsilon and without assuming that the limit of a sum of two functions is the sum of their individual limits) that the limx-->a[x+f(x)]=a+L ?

Thanks for the help!
 
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  • #2
(using delta-epsilon and without assuming that the limit of a sum of two functions is the sum of their individual limits)

You can always use the same proof sketch of limit arithmetics, only specifically applied to your case ;)
 
  • #3
Hint: use ε/2.
 

Related to Proving with Delta Epsilon: A Beginner's Guide

1. What is Delta Epsilon and why is it important in proving?

Delta Epsilon is a mathematical concept used in the process of proving. It involves using a small value, represented by the symbol delta, to show that a function approaches a certain value, represented by the symbol epsilon. This is important because it allows us to show that a function is continuous and has a limit at a certain point.

2. How do I use Delta Epsilon to prove a function is continuous?

To use Delta Epsilon in proving continuity, you must show that for any given epsilon, there exists a delta such that if the distance between the input and the point of interest is less than delta, the output will be within epsilon of the limit. This shows that the function is approaching the limit as the input gets closer to the point of interest.

3. Can Delta Epsilon be used to prove limits at all points?

No, Delta Epsilon can only be used to prove limits at points where the function is continuous. If a function is not continuous at a point, Delta Epsilon cannot be used to prove the limit at that point.

4. What is the significance of using a small value for delta?

The small value of delta ensures that the function is approaching the limit at a certain point, rather than just staying within a certain range. It also allows for greater precision in proving the continuity of a function.

5. Are there any common errors to avoid when using Delta Epsilon in proving?

One common error is using a delta that is too large, which can result in the function not approaching the limit as desired. It is important to choose a small enough delta that will accurately show the behavior of the function near the point of interest. It is also important to clearly define the function, the point of interest, and the desired limit to avoid any confusion or mistakes in the proof.

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