# 极限 ## 1 定义 $$ \begin{aligned} & {C}&{\text{常}\text{数}}\\ & {f{ ( {x}) }}&{\text{函}\text{数}}\\ & {n}&{\text{正}\text{整}\text{数}}\\ &{ \{ {\mathop{{x}}\nolimits_{{n}}} \} ,{ \{ {\mathop{{y}}\nolimits_{{n}}} \} } }&{\text{数}\text{列}} \\ \end{aligned} $$ $$ \begin{aligned} & { \text{lim} { \left[ {Cf{ \left( {x} \right) }} \left] =C{ \left[ { \text{lim} f{ \left( {x} \right) }} \right] }\right. \right. }}\\ & { \text{lim} {\mathop{{ \left[ {f{ \left( {x} \right) }} \right] }}\nolimits^{{n}}}={\mathop{{ \left[ { \text{lim} f{ \left( {x} \right) }} \right] }}\nolimits^{{n}}}}\\ & {\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}{ \left( {\mathop{{x}}\nolimits_{{n}} \pm \mathop{{y}}\nolimits_{{n}}} \right) }=\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{x}}\nolimits_{{n}} \pm \mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{y}}\nolimits_{{n}}}\\ & {\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}{ \left( {\mathop{{x}}\nolimits_{{n}} \cdot \mathop{{y}}\nolimits_{{n}}} \right) }=\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{x}}\nolimits_{{n}} \cdot \mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{y}}\nolimits_{{n}}}\\ &{\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}{ \left( {\frac{{\mathop{{x}}\nolimits_{{n}}}}{{\mathop{{y}}\nolimits_{{n}}}}} \right) }=\frac{{\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{x}}\nolimits_{{n}}}}{{\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{y}}\nolimits_{{n}}}},{ \left( {\mathop{{y}}\nolimits_{{n}} \neq 0,n=1,2,3, \cdots } \right) }} \end{aligned} $$ ## 2 公式 $$ \begin{aligned} \end{aligned} $$ ## 3 定理 ### 洛必达法则1 $$ \begin{aligned} & {\text{若}\text{函}\text{数}f{ \left( {x} \left) ,g{ \left( {x} \right) }\text{满}\text{足}\text{如}\text{下}\text{条}\text{件}\text{:}\right. \right. }}\\ & {\mathop{{ \text{lim} }}\limits_{{x \to a}}f{ \left( {x} \right) }=0,\mathop{{ \text{lim} }}\limits_{{x \to a}}g{ \left( {x} \right) }=0}\\ & {\text{在}\text{点}a\text{的}\text{某}\text{去}\text{邻}\text{域}\text{内}f{ \left( {x} \left) \text{和}g{ \left( {x} \right) }\text{都}\text{可}\text{导}\text{,}\text{且}{g} \prime { \left( {x} \left) \neq 0\right. \right. }\right. \right. }}\\ & {\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f \prime { \left( {x} \right) }}}{{g \prime { \left( {x} \right) }}}=A,{ \left( {A\text{为}\text{实}\text{数}} \right) }}\\ & {\text{则}\text{有}}\\ & {\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f{ \left( {x} \right) }}}{{g{ \left( {x} \right) }}}=\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f \prime { \left( {x} \right) }}}{{g \prime { \left( {x} \right) }}}=A} \end{aligned} $$ ### 洛必达法则2 $$ \begin{aligned} & {\text{若}\text{函}\text{数}f{ \left( {x} \left) ,g{ \left( {x} \right) }\text{满}\text{足}\text{如}\text{下}\text{条}\text{件}\text{:}\right. \right. }}\\ & {\mathop{{ \text{lim} }}\limits_{{x \to a}}f{ \left( {x} \right) }=0,\mathop{{ \text{lim} }}\limits_{{x \to a}}g{ \left( {x} \right) }=0}\\ & {\text{在}\text{点}a\text{的}\text{某}\text{去}\text{邻}\text{域}\text{内}f{ \left( {x} \left) \text{和}g{ \left( {x} \right) }\text{都}\text{可}\text{导}\text{,}\text{且}{g} \prime { \left( {x} \left) \neq 0\right. \right. }\right. \right. }}\\ & {\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f \prime { \left( {x} \right) }}}{{g \prime { \left( {x} \right) }}}=A,{ \left( {A\text{为}\text{实}\text{数}} \right) }}\\ & {\text{则}\text{有}}\\ & {\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f{ \left( {x} \right) }}}{{g{ \left( {x} \right) }}}=\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f \prime { \left( {x} \right) }}}{{g \prime { \left( {x} \right) }}}=A} \end{aligned} $$
