$\bf命题：$设$\int_a^{ + \infty } {f\left( x \right)dx} $收敛，若$\lim \limits_{x \to \begin{array}{*{20}{c}} {{\rm{ + }}\infty } \end{array}
<script id="MathJax-Element-1" type="math/tex; mode=display">\begin{array}{*{20}{c}} {{\rm{ + }}\infty } \end{array}</script>} f\left( x \right)$存在，则$\lim \limits_{x \to \begin{array}{*{20}{c}} {{\rm{ + }}\infty } \end{array}
<script id="MathJax-Element-2" type="math/tex; mode=display">\begin{array}{*{20}{c}} {{\rm{ + }}\infty } \end{array}</script>} f\left( x \right) = 0$参考答案
$\bf命题：$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$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}
<script id="MathJax-Element-3" type="math/tex; mode=display">\begin{array}{*{20}{c}}{ + \infty }\end{array}</script>} f\left( x \right) = 0$参考答案
$\bf命题：$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}
<script id="MathJax-Element-4" type="math/tex; mode=display">\begin{array}{*{20}{c}}{ + \infty }\end{array}</script>} xf\left( x \right) = 0$，进而$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}
<script id="MathJax-Element-5" type="math/tex; mode=display">\begin{array}{*{20}{c}}{ + \infty }\end{array}</script>} f\left( x \right) = 0$参考答案
$\bf命题：$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
参考答案
$\bf命题：$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}
<script id="MathJax-Element-6" type="math/tex; mode=display">\begin{array}{*{20}{c}} { + \infty } \end{array}</script>} xf\left( x \right) = 0$参考答案
$\bf命题：$设$f\left( x \right)$单调且$\lim \limits_{x \to \begin{array}{*{20}{c}} {{0^ + }} \end{array}
<script id="MathJax-Element-7" type="math/tex; mode=display">\begin{array}{*{20}{c}} {{0^ + }} \end{array}</script>} 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}
<script id="MathJax-Element-8" type="math/tex; mode=display">\begin{array}{*{20}{c}} {{0^ + }} \end{array}</script>} xf\left( x \right) = 0$参考答案
$\bf命题：$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}
<script id="MathJax-Element-9" type="math/tex; mode=display">\begin{array}{*{20}{c}} { + \infty } \end{array}</script>} xf\left( x \right)\ln x = 0$参考答案
$\bf命题：$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}
<script id="MathJax-Element-10" type="math/tex; mode=display">\begin{array}{*{20}{c}}{ + \infty }\end{array}</script>} f\left( x \right) = 0$方法一 方法二
$\bf命题：$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}
<script id="MathJax-Element-11" type="math/tex; mode=display">\begin{array}{*{20}{c}} { + \infty } \end{array}</script>} f\left( x \right) = 0$$\bf命题：$设$f\left( x \right)$在${\left[ {a, + \infty } \right)}$上可导且导函数有界，若$\int_a^{ + \infty } {f\left( x \right)dx} $绝对收敛，则$\lim \limits_{x \to \begin{array}{*{20}{c}} { + \infty } \end{array}
<script id="MathJax-Element-12" type="math/tex; mode=display">\begin{array}{*{20}{c}} { + \infty } \end{array}</script>} f\left( x \right) = 0$参考答案
$\bf命题：$设$f\left( x \right)$在${\left[ {a, + \infty } \right)}$上可导且导函数有界，若$ \int_a^{ + \infty } {{f^2}\left( x \right)dx} < + \infty $，则$\lim \limits_{x \to \begin{array}{*{20}{c}} { + \infty } \end{array}
<script id="MathJax-Element-13" type="math/tex; mode=display">\begin{array}{*{20}{c}} { + \infty } \end{array}</script>} f\left( x \right) = 0$$\bf命题：$设$p \ge 1,f\left( x \right) \in {C^1}\left( { - \infty , + \infty } \right)$，且\int_{ - \infty }^{ + \infty } {{{\left| {f\left( x \right)} \right|}^p}dx} < + \infty ,\int_{ - \infty }^{ + \infty } {{{\left| {f‘\left( x \right)} \right|}^p}dx} < + \infty
<script id="MathJax-Element-14" type="math/tex; mode=display">\int_{ - \infty }^{ + \infty } {{{\left| {f\left( x \right)} \right|}^p}dx} < + \infty ,\int_{ - \infty }^{ + \infty } {{{\left| {f‘\left( x \right)} \right|}^p}dx} < + \infty </script>
证明：$\lim \limits_{x \to \begin{array}{*{20}{c}}\infty \end{array}
<script id="MathJax-Element-15" type="math/tex; mode=display">\begin{array}{*{20}{c}}\infty \end{array}</script>} 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}
<script id="MathJax-Element-16" type="math/tex; mode=display">{\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}</script>参考答案
$\bf命题：$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
<script id="MathJax-Element-17" type="math/tex; mode=display">\mathop {\lim }\limits_{n \to\infty } {x_n} = + \infty ，\mathop {\lim }\limits_{n \to \infty } f\left( {{x_n}} \right) = 0</script>参考答案
$\bf命题：$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}
<script id="MathJax-Element-18" type="math/tex; mode=display">\begin{array}{*{20}{c}}{{\rm{ + }}\infty }\end{array}</script>} f\left( x \right) = 0$，则$\int_a^{{\rm{ + }}\infty } {{f^2}\left( x \right)dx} $收敛 参考答案
$\bf命题：$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
参考答案
$\bf命题：$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}
<script id="MathJax-Element-19" type="math/tex; mode=display">\begin{array}{*{20}{c}}{ + \infty }\end{array}</script>} f‘‘\left( x \right) = + \infty $，则$\int_1^{ + \infty } {\frac{1}{{f\left( x \right)}}dx} $收敛参考答案
$\bf命题：$
