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Let f(t) be an integrable function in any interval.

Let h(t) be defined on [0, oo[ such that

h(0) = 0

and for any other "t", h(t) = (1 - cos(At)) / t

It is not difficult to see that h is integrable on [0, b] for any positive "b", so fh is also integrable in said interval.

Considering fh as an integrand, Let the function G(b) be defined in the domain [0, oo] and = to the definite integral from 0 to b.

How to proof that lim G(b) exists? (b --> oo)

(In pg. 472 of first edition of Mathematical Analysis Apostol says that it indeed does so).

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