Jack said:Suppose {f_n} is a sequence of functions that converges almost everywhere to a function f
and define F_n = sup_k=1,...n |f_n| .
Show that if the integrals of F_n remain bounded as n goes to infinity,
then lim_n ∫〖f_n dm〗 = ∫〖f dm〗.
The definition of a limit in calculus is the value that a function approaches as the input variable gets closer and closer to a certain value or point. It is denoted by the notation "lim".
In order to show that limn→∞∫n 〖fn dm〗 = ∫〖f dm〗, you must first show that the limit of fn as n approaches infinity exists. Then, you can use the definition of a limit to show that the limit of the integral of fn is equal to the integral of the limit of fn.
The conditions for the equality of the limit of an integral and the integral of the limit are that the limit of the function must exist, the function must be continuous, and the domain of the function must be closed and bounded.
No, the limit of an integral can only be equal to the integral of the same function. This is because the definition of a limit requires the function to approach the same value as the input variable approaches a certain point.
Yes, there are special cases where the limit of an integral is not equal to the integral of the limit. These cases include when the function is discontinuous, when the domain is not closed and bounded, and when the limit of the function does not exist.