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Fie \( f:[0,\infty)\to\mathbb{R} \) o functie continua astfel incat

\( \lim_{x\to\infty}\left(f(x)+\int_0^xf(t)dt\right) \) exista.

Aratati ca \( \lim_{x\to\infty}f(x)=0 \).

\( F(x)=\int_{0}^{x}f(t)dt \)

\( F^{\prime}(x)+F(x)=g(x) \) tends to \( L\in R \)

Solving the differential equation
\( F(x)=e^{-x}\int_{0}^{x}e^{t}g(t)dt+Ae^{-x} \)
with A fixed real, we get

\( f(x)=F^{\prime}(x)=-\frac{\int_{0}^{x}e^{t}g(t)dt}{e^{x}}+g(x)-Ae^{-x} \)

Now if \( g \) tends to \( L \), then \( \frac{\int_{0}^{x}e^{t}g(t)dt}{e^{x}} \) tends also to \( L \).

Finally \( f \) tends to \( 0 \).