$\bf命题：$设$f\left( x \right) \in C\left[ {0,1} \right]$，证明：$\lim \limits_{h \to \begin{array}{*{20}{c}}{{0^ + }} \end{array}} \int_0^1 {\frac{h}{{{x^2} + {h^2}}}} f\left( x \right)dx = \frac{\pi }{2}f\left( 0 \right)$
$\bf命题：$设$\forall [\alpha ,\beta ] \subset \left[ {0, + \infty } \right),f\left( x \right) \in R[\alpha ,\beta ]$，且$\int_0^{ + \infty } {f\left( x \right)dx} $收敛，则对于常数$a>1$成立\[\mathop {\lim }\limits_{y \to \begin{array}{*{20}{c}}{{0^ + }} \end{array}} \int_0^{ + \infty } {{a^{ - xy}}f\left( x \right)dx} = \int_0^{ + \infty } {f\left( x \right)dx} \]
1
$\bf命题：$设$\forall a \in \left( {0, + \infty } \right),f\left( x \right) \in R\left[ {0,a} \right]$，且$\mathop {\lim }\limits_{x \to + \infty } f\left( x \right) = \alpha $，则
\[\mathop {\lim }\limits_{t \to \begin{array}{*{20}{c}}
{{0^ + }}
\end{array}} t\int_0^{ + \infty } {{e^{ - tx}}f\left( x \right)dx} = \alpha \]
$\bf命题：$设$\forall \left[ {a,b} \right] \subset \left( { - \infty , + \infty } \right),f\left( x \right) \in R\left[ {a,b} \right]$，且$\int_{ - \infty }^{ + \infty } {\left| {f\left( x \right)} \right|dx} $，则$F\left( x \right) = \int_{ - \infty }^{ + \infty } {f\left( x \right)\cos \left( {2\pi tx} \right)dt} $在$\left( { - \infty , + \infty } \right)$上一致连续
$\bf命题：$设$f\left( x \right) \in C\left[ {0, + \infty } \right)$，且$\int_0^{ + \infty } {\varphi \left( x \right)dx} $绝对收敛，则\[\mathop {\lim }\limits_{n \to \infty } \int_0^{\sqrt n } {f\left( {\frac{x}{n}} \right)\varphi \left( x \right)dx} = f\left( 0 \right)\int_0^{ + \infty } {\varphi \left( x \right)dx} \]
$\bf命题：$
附录
$\bf（Tauber定理）$设幂级数$\sum\limits_{n = 0}^\infty {{a_n}{x^n}} = S\left( x \right)$在$\left( { - 1,1} \right)$上成立，若$\lim \limits_{x \to \begin{array}{*{20}{c}}{{1^ - }}\end{array}} S\left( x \right) = S$，且$\lim \limits_{n \to \infty } n{a_n} = 0$，则$\sum\limits_{n = 0}^\infty {{a_n}} = S$
$\bf（Riemman积分控制收敛定理）$设函数列$\left\{ {{f_n}\left( x \right)} \right\}$在$\left[ {a, + \infty } \right)$上内闭可积且一致收敛于$f(x)$，若存在函数$g(x)$，使得$\left| {{f_n}\left( x \right)} \right| \le g\left( x \right)$对每个$x$与$n$都成立，且$\int_a^{ + \infty } {g\left( x \right)dx} $收敛，则\[\mathop {\lim }\limits_{n \to \infty } \int_a^{ + \infty } {{f_n}\left( x \right)dx} {\rm{ = }}\int_a^{ + \infty } {f\left( x \right)dx} \]