How shall we show that this limit exists?

In summary, we have an integral expression for ##f## involving the distance between two points and their coordinates. To prove the existence of the limit, we need to show that we can take the limit inside the integral, which is where the difficulty lies. Further assistance is needed to proceed with the proof.
  • #1
Mike400
59
6
Moved from a technical forum
Let:

##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'##

where ##V'## is a finite volume in space
##\mathbf{r}=(x,y,z)## are coordinates of all space
##\mathbf{r'}=(x',y',z')## are coordinates of ##V'##
##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##

How to prove that:

##\lim\limits_{\Delta x \to 0} \dfrac{f(x+\Delta x,y,z)-f(x,y,z)}{\Delta x}## exist
 
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  • #2
What have you attempted to find the solution?
 
  • #3
DrClaude said:
What have you attempted to find the solution?
##\lim\limits_{\Delta x \to 0} \dfrac{f(x+\Delta x,y,z)-f(x,y,z)}{\Delta x}\\
=\lim\limits_{\Delta x \to 0}\dfrac{\displaystyle\int_{V'} \dfrac{(x+\Delta x)-x'}{|\mathbf{r}(x+\Delta x,y,z)-\mathbf{r'}|^3}\ dV' - \int_{V'} \dfrac{x-x'}{|\mathbf{r}(x,y,z)-\mathbf{r'}|^3}\ dV'}{\Delta x}\\
=\lim\limits_{\Delta x \to 0}\displaystyle\int_{V'}
\dfrac{\left( \dfrac{(x+\Delta x)-x'}{|\mathbf{r}(x+\Delta x,y,z)-\mathbf{r'}|^3}
-\dfrac{x-x'}{|\mathbf{r}(x,y,z)-\mathbf{r'}|^3} \right)}{\Delta x}dV'##

Now if only I could take the limit inside the integral (with respect to ##V'##), I can proceed to show the limit exists. One of my colleagues told me that we cannot always take such limits inside the integral. This is where I am stuck. Hope someone here can help. Thanks in advance.
 

FAQ: How shall we show that this limit exists?

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 represents the value that a function approaches as its input gets closer and closer to a specific value, but may never actually reach that value.

2. How do we determine if a limit exists?

To determine if a limit exists, we must evaluate the function at values approaching the given input from both the left and right sides. If the values approach the same number, the limit exists. If the values approach different numbers or the function is undefined at the given input, the limit does not exist.

3. Can a limit exist even if the function is not defined at the given input?

Yes, a limit can exist even if the function is not defined at the given input. This is because a limit only considers the behavior of the function as the input approaches the given value, not necessarily the actual value of the function at that point.

4. What are the different types of limits?

The different types of limits are finite limits, infinite limits, and limits at infinity. Finite limits exist when the function approaches a specific value as the input approaches a given value. Infinite limits occur when the function approaches positive or negative infinity as the input approaches a given value. Limits at infinity describe the behavior of a function as the input approaches positive or negative infinity.

5. Why is it important to determine if a limit exists?

Determining if a limit exists is important because it helps us understand the behavior of a function and make predictions about its values. It is also a crucial step in solving more complex mathematical problems and proving theorems in calculus and other areas of mathematics.

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