Commute of Limits and Integrals: Understanding the Relationship

[moo5003]

I was wondering if Limits commute,

Ie:
Limit of z to infinity[ Limit of z to 0[ f(0)/z ]]

=

Limit of z to 0[ Limit of z to infinity[ f(0)/z ]]

Edit: Nvm, they dont... I just did a some test functions... I'm just hoping for an easy way to finish one of my proofs.

Though on a sidenote, integrals and limits do commute right?

 

[AKG]

This doesn't even make sense. The only way "Limit of z to A[Expression]" even makes sense is if z is a free variable in the Expression. But in both "Limit of z to 0[ f(0)/z ]" and "Limit of z to infinity[ f(0)/z ]", z is NOT a free variable. Note that in "f(0)/z", z is a free variable, but in "Limit of z to B[ f(0)/z ]", it is not. (In the above, A and B can of course be anything, not just 0 and infinity which you've used in your examples). So I don't see how you could have determined:

Edit: Nvm, they dont... I just did a some test functions...

since it doesn't even make sense.

And integrals and limits don't always commute.

 

[tacman]

Yes, integrals and limits do commute. This means that the order in which you take the integral and the limit does not affect the final result. This is known as the Limit of Integrals Theorem. However, there are certain conditions that need to be met for this to hold true, such as the function being continuous and the integral being finite.

In terms of your initial question about limits commuting, it is important to understand the relationship between limits and integrals. Limits are used to find the behavior of a function as a variable approaches a specific value, while integrals are used to find the area under a curve. These concepts are closely related and often used together in calculus.

In the example you provided, the limits do not commute because the order in which they are taken affects the final result. This is because the limit as z approaches infinity will yield a different value than the limit as z approaches 0. Therefore, it is important to be aware of the order in which you take limits, as it can greatly impact the final result.

In terms of your proof, it is important to carefully consider the order in which you take limits and ensure that all conditions for the Limit of Integrals Theorem are met. This will help you to successfully complete your proof. Additionally, it may be helpful to review the properties of limits and integrals to gain a better understanding of their relationship and how they can be manipulated.