Switching Limit & Integral in Limits & Integrals

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The discussion centers on the interchangeability of limits and integrals, specifically when evaluating the limit of an integral involving separate variables. The example provided demonstrates that the limit as z approaches a of the integral of the function f(x)/((x-z)(x-a)^2) can be transformed into the integral of the limit of the same function, provided that both the limit and integral are uniformly convergent. The conclusion emphasizes that this interchange is valid as long as there are no infinite or indeterminate forms present.

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If you are taking the limit of an integral, can you switch the composition ie: take the integral of a limit if the limit and integral are on separate variables?

Ie:

Lim of z to a [integral over alpha [f(x)/((x-z)(x-a)^2) dx]

=

Integral over alpha[Lim of z to a[f(x)/((x-z)(x-a)^2) dx]

=

Integral over alpha[f(x)/(x-a)^3 dx]
 
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Since an integration involves limits, and switching the order of taking limits is a tricky business, i'd say that you could do the permutation, as long as there's no infinity (or no indeterminate expression under the limit sign) involved...My guess...
 
Generally speaking, if both limit and integral are "uniformly convergent" then they can be interchanged.
 

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