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.