Are Extra Conditions Affecting the Limit Definition?

• I
These choices could potentially cause confusion and should be revised. Overall, the author's argument is valid and the conclusion is supported by the given information.
TL;DR Summary
Do the following extra highlighted words in the ##\epsilon-\delta## definition of limit prevent us from concluding that the limit exists? Why? Why not?
This question consists of two parts: preliminary and the main question. Reading only the main question may be enough to get my point, but if you want details please have a look at the preliminary.

PRELIMINARY:

Let potential due to a small volume ##\delta## at a point ##(1,2,3)## inside it be denoted by ##\psi_{\delta}##.

_______________________________________________________________________________________________________________________________________________________________________

It can be shown that for every ##\epsilon>0##, we can choose a volume ##\delta## such that:

##\left| \dfrac{\psi_{\delta}(1+\Delta x,2,3)-\psi_{\delta}(1,2,3)}{\Delta x} \right| < \dfrac{\epsilon}{3} \tag1##

That is ##\epsilon## can be made as small as we can by choosing a small volume ##\delta##.
_______________________________________________________________________________________________________________________________________________________________________

Also, using spherical coordinate system we can show that for every ##\epsilon>0##, we can choose a volume ##\delta## such that:

##\displaystyle\left| \iiint_{\delta} \dfrac{\rho'}{R^2} \dfrac{x-x'}{R} dV' \right| < \dfrac{\epsilon}{3}##

That is:

##\displaystyle\left| \iiint_{V'} \dfrac{\rho'}{R^2} \dfrac{x-x'}{R} dV'
-\iiint_{(V'-\delta)} \dfrac{\rho'}{R^2} \dfrac{x-x'}{R} dV' \right| < \dfrac{\epsilon}{3} \tag2 ##

That is ##\epsilon## can be made as small as we can by choosing a small volume ##\delta##.
_______________________________________________________________________________________________________________________________________________________________________

No matter what our volume ##\delta## is, at point ##P(1,2,3)##, ##\dfrac{\partial \psi_{(V'-\delta)}}{\partial x}## exists (since ##P## being an outside point of ##V'−\delta##). That is:

##\lim\limits_{\Delta x \to 0} \dfrac{\psi_{(V'-\delta)}(1+\Delta x,2,3)-\psi_{(V'-\delta)}(1,2,3)}{\Delta x}=\dfrac{\partial \psi_{(V'-\delta)}}{\partial x} (1,2,3)##

That is, for every ##\epsilon>0##, we can choose an interval ##\delta x## around ##\Delta x=0## (inside volume ##\delta##) such that whenever ##0<|\Delta x−0|<\delta x##:

##\left| \dfrac{\psi_{(V'-\delta)}(1+\Delta x,2,3)-\psi_{(V'-\delta)}(1,2,3)}{\Delta x} - \dfrac{\partial \psi_{(V'-\delta)}}{\partial x} (1,2,3) \right| < \dfrac{\epsilon}{3}##

That is:
##\left| \dfrac{\psi_{(V'-\delta)}(1+\Delta x,2,3)-\psi_{(V'-\delta)}(1,2,3)}{\Delta x} - \left( -\displaystyle\iiint_{(V'-\delta)} \dfrac{\rho'}{R^2} \dfrac{x-x'}{R} dV' \right) \right| < \dfrac{\epsilon}{3} \tag3##

That is ##\epsilon## can be made as small as we can by choosing a small interval ##\delta x## around ##\Delta x=0## (inside volume ##\delta##)
_______________________________________________________________________________________________________________________________________________________________________

##\left| \dfrac{\psi_{V'}(1+\Delta x,2,3)-\psi_{V'}(1,2,3)}{\Delta x} - \left( -\displaystyle \iiint_{V'} \dfrac{\rho'}{R^2} \dfrac{x-x'}{R} dV' \right) \right| < \epsilon##

That is, for every ##\epsilon>0##, we can choose ##\bbox[yellow]{\text{a volume δ and}}## an interval ##\delta x## around ##\Delta x=0## (inside volume ##\delta##) such that whenever ##0<|\Delta x−0|<\delta x##, the above inequality holds.

That is, ##\epsilon## can be made as small as we can by choosing ##\bbox[yellow]{\text{a volume δ and}}## a small interval ##\delta x## around ##\Delta x=0## (inside volume ##\delta##)

QUESTION:

Since there are some extra highlighted words in the above ##\epsilon-\delta## definition of limit, will this prevent us from saying that:

##\lim\limits_{\Delta x \to 0} \dfrac{\psi_{V'}(1+\Delta x,2,3)-\psi_{V'}(1,2,3)}{\Delta x}=-\displaystyle \iiint_{V'} \dfrac{\rho'}{R^2} \dfrac{x-x'}{R} dV'##

? Why? Why not?

berkeman
Since there are some extra highlighted words in the above ##\epsilon-\delta## definition of limit, will this prevent us from saying that:
##\lim\limits_{\Delta x \to 0} \dfrac{\psi_{V'}(1+\Delta x,2,3)-\psi_{V'}(1,2,3)}{\Delta x}=-\displaystyle \iiint_{V'} \dfrac{\rho'}{R^2} \dfrac{x-x'}{R} dV'##
This seems to be a reasonable conclusion to me.
However, the author's choice of ##\delta## to represent a volume seems very ill-chosen to me, as well as ##\delta x## for an interval around ##\Delta x = 0##.

1. What is a limit in mathematics?

A limit in mathematics is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It is used to describe the value that a function approaches, rather than the value it actually reaches.

2. How do you determine if a limit exists?

A limit exists if the value that the function approaches from the left side is equal to the value it approaches from the right side. In other words, the left and right limits must be equal for the overall limit to exist.

3. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of a function as its input approaches a certain value from one direction (either the left or the right). A two-sided limit considers the behavior from both directions and requires that the left and right limits are equal for the overall limit to exist.

4. Can a limit be undefined?

Yes, a limit can be undefined if the left and right limits approach different values or if one or both of the limits do not exist. In this case, we say that the limit does not exist.

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

Limits are used in many real-world applications, such as calculating the speed of an object at a specific time or determining the average rate of change in a process. They are also used in physics, engineering, and economics to model and predict the behavior of systems.

Replies
10
Views
1K
Replies
4
Views
1K
Replies
14
Views
2K
Replies
5
Views
1K
Replies
4
Views
2K
Replies
1
Views
2K
Replies
25
Views
3K
Replies
16
Views
3K
Replies
2
Views
1K
Replies
9
Views
1K