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## Main Question or Discussion Point

Let A be a constant.

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).

Sorry for not using latex but there is some technical problem in some server...

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).

Sorry for not using latex but there is some technical problem in some server...