- #1
geoffrey159
- 535
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Homework Statement
Let ##f## be piecewise continuous from ##[0,+\infty[## into ##V = \mathbb{R} ## or ##\mathbb{C}##, such that ## f(x) \longrightarrow_{ x\rightarrow +\infty} \ell ##.
Show that ## \frac{1}{x}\ \int_0^x f(t) \ dt \longrightarrow_{ x\rightarrow +\infty} \ell##
Homework Equations
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Integration of asymptotic comparisons
The Attempt at a Solution
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Can you tell me if this is correct please ?
Since ##f - \ell = o_{+\infty}(1) ##, and since ## u: x \rightarrow 1 ## is non-negative, piecewise continuous, and non-integrable on ##[0,+\infty[##, then
## \int_0^x f(t) - \ell \ dt = o_{+\infty}(\int_0^x u(t) \ dt)##
which is the same as saying that ##\int_0^x f(t) \ dt - x \ell = o_{+\infty}(x) ##.
Multiplying left and right by ##\frac{1}{x}##, I get that ## \frac{1}{x}\ \int_0^x f(t) \ dt - \ell = o_{+\infty}(1)## which proves that
## \frac{1}{x}\ \int_0^x f(t) \ dt \longrightarrow_{ x\rightarrow +\infty} \ell##.
Is this OK ?