(1) lim ⁡ x → a f ( x ) = lim ⁡ x → a g ( x ) = 0 \lim \limits_{x \rightarrow a} f(x)=\lim \limits_{x \rightarrow a} g(x)=0 x→alim​f(x)=x→alim​g(x)=0 ​​​

(2) f ( x ) , g ( x ) f(x), g(x) f(x),g(x)​​​ 在 a a a​​​ 点邻域可导，且 g ′ ( x ) ≠ 0 g^{\prime}(x) \neq 0 g′(x)=0 ​​​

(3) lim ⁡ x → a f ′ ( x ) g ′ ( x ) = A ( A \lim \limits_{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}=A(A x→alim​g′(x)f′(x)​=A(A​​​ 可以为 ∞ ) \infty) ∞) ​​​

⇒ lim ⁡ x → a f ( x ) g ( x ) = A \Rightarrow \quad \lim \limits_{x \rightarrow a} \frac{f(x)}{g(x)}=A ⇒x→alim​g(x)f(x)​=A ​

即 lim ⁡ x → a f ( x ) g ( x ) = lim ⁡ x → a f ′ ( x ) g ′ ( x ) \lim \limits_{x \rightarrow a} \frac{f(x)}{g(x)}=\lim \limits_{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)} x→alim​g(x)f(x)​=x→alim​g′(x)f′(x)​​ （在右端有意义的情况下成立）