$\bf命题：$设$\int_a^{ + \infty } {f\left( x \right)dx}
$收敛，若$\lim \limits_{x \to \begin{array}{*{20}{c}}
{{\rm{ + }}\infty } \end{array}
+∞
} f\left( x \right)$存在，则$\lim \limits_{x \to \begin{array}{*{20}{c}} {{\rm{ + }}\infty }
\end{array}
} f\left( x \right) = 0$
$\bf命题：$设$f\left( x \right) \in {C^1}\left[ {a, + \infty }
\right)$，若$\int_a^{ + \infty } {f\left( x \right)dx} ，\int_a^{ + \infty }
{f‘\left( x \right)dx}$均收敛，则$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}
} f\left( x \right) = 0$
$\bf命题：$设${f\left( x \right)}$在$\left[ {a,{\rm{ + }}\infty }
\right)$单调，且$\int_a^{ + \infty } {f\left( x \right)dx} $收敛，则$\lim \limits_{x \to
\begin{array}{*{20}{c}}{ + \infty
}\end{array}
} xf\left( x \right) = 0$，进而$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}
} f\left( x \right) = 0$
$\bf命题：$设${f\left( x \right)}$在$\left[ {a, + \infty }
\right)$上可微且单调下降，若$\int_a^{ + \infty } {f\left( x \right)dx} $收敛，则$\int_a^{ +
\infty } {xf‘\left( x \right)dx} $收敛
$\bf命题：$设$\int_a^{ + \infty } {f\left( x \right)dx} $收敛，且$\frac{{f\left( x
\right)}}{x}$在${\left[ {a, + \infty } \right)}$上单调递减，则$\lim \limits_{x \to \begin{array}{*{20}{c}} { + \infty } \end{array}
} xf\left( x \right) = 0$
$\bf命题：$设$f\left( x \right)$单调且$\lim \limits_{x \to \begin{array}{*{20}{c}}
{{0^ + }} \end{array}
} f\left( x \right) = + \infty $，若$\int_0^1 {f\left( x \right)dx} $收敛，则$\lim
\limits_{x \to \begin{array}{*{20}{c}} {{0^ + }}
\end{array}
} xf\left( x \right) = 0$
$\bf命题：$设$xf\left( x \right)$在${\left[ {a, + \infty }
\right)}$上单调递减，若$\int_a^{ + \infty } {f\left( x \right)dx} $收敛，则$\lim \limits_{x
\to \begin{array}{*{20}{c}} { + \infty }
\end{array}
} xf\left( x \right)\ln x = 0$
$\bf命题：$设$f\left( x \right)$在${\left[ {a, + \infty }
\right)}$上一致连续，若$\int_a^{ + \infty } {f\left( x \right)dx} $收敛，则$\lim \limits_{x
\to \begin{array}{*{20}{c}}{ + \infty
}\end{array}
} f\left( x \right) = 0$
$\bf命题：$设$f\left( x \right)$在${\left[ {a, + \infty }
\right)}$上可导且导函数有界，若$\int_a^{ + \infty } {f\left( x \right)dx} $收敛，则$\lim
\limits_{x \to \begin{array}{*{20}{c}} { + \infty
} \end{array}
} f\left( x \right) = 0$
$\bf命题：$设$f\left( x \right)$在${\left[ {a, + \infty }
\right)}$上可导且导函数有界，若$\int_a^{ + \infty } {f\left( x \right)dx} $绝对收敛，则$\lim
\limits_{x \to \begin{array}{*{20}{c}} { + \infty
} \end{array}
} f\left( x \right) = 0$
$\bf命题：$设$f\left( x \right)$在${\left[ {a, + \infty } \right)}$上可导且导函数有界，若$
\int_a^{ + \infty } {{f^2}\left( x \right)dx} < + \infty $，则$\lim
\limits_{x \to \begin{array}{*{20}{c}} { + \infty
} \end{array}
} f\left( x \right) = 0$
$\bf命题：$设$p \ge 1,f\left( x \right) \in {C^1}\left( { - \infty , + \infty }
\right)$，且\int_{ - \infty }^{ + \infty } {{{\left|
{f\left( x \right)} \right|}^p}dx} < + \infty ,\int_{ - \infty }^{ + \infty
} {{{\left| {f‘\left( x \right)} \right|}^p}dx} < + \infty
证明：$\lim \limits_{x \to \begin{array}{*{20}{c}}\infty \end{array}
} f\left( x \right) = 0$，且{\left| {f\left( x
\right)} \right|^p} \le \frac{{p - 1}}{2}\int_{ - \infty }^{ + \infty }
{{{\left| {f\left( t \right)} \right|}^p}dt} + \frac{1}{2}\int_{ - \infty }^{ +
\infty } {{{\left| {f‘\left( t \right)} \right|}^p}dt}
$\bf命题：$设$f\left( x \right) \in C\left[ {a, + \infty } \right)$，且$\int_a^{ +
\infty } {f\left( x \right)dx} $收敛，则存在数列$\left\{ {{x_n}} \right\} \subset \left[
{a, + \infty } \right)$，使得\mathop {\lim
}\limits_{n \to\infty } {x_n} = + \infty ，\mathop {\lim }\limits_{n \to \infty }
f\left( {{x_n}} \right) = 0
$\bf命题：$设$\int_a^{{\rm{ + }}\infty } {f\left( x \right)dx} $绝对收敛，且$\lim
\limits_{x \to \begin{array}{*{20}{c}}{{\rm{ +
}}\infty }\end{array}
} f\left( x \right) = 0$，则$\int_a^{{\rm{ + }}\infty } {{f^2}\left( x \right)dx}
$收敛
$\bf命题：$设$f\left( x \right)$在$\left[ {0, + \infty }
\right)$上可微，$f‘\left( x \right)$在$\left[ {0, + \infty }
\right)$上单调递增且无上界，则$\int_0^{ + \infty } {\frac{1}{{1 + {f^2}\left( x
\right)}}dx} $收敛
$\bf命题：$设正值函数$f\left( x \right)$在$\left[ {1, +
\infty } \right)$上二阶连续可微，且$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}
} f‘‘\left( x \right) = + \infty $，则$\int_1^{ + \infty } {\frac{1}{{f\left(
x \right)}}dx} $收敛
$\bf命题：$