定理如果函数f(x)和g(x满足如下条件: (1)f(x)和g(x)都是当xx时的无穷小(或无穷大); (2)fx)和g(x)在点a的某去心邻域内都可导且g(x)≠0; f() (3)lim 存在(或为无穷大) x→>ag(x) f(x 那 lim f'(x) Im x→a g(x) x> g(x) 简要证明令fa)=g(a)=0,于是f(x)及g(x)在点a的某邻域 内连续.在该邻域内应用柯西中值定理,有 lim f(r) 2)=lim /(x)-f(a)=lim /(5) xa g(x) x>a g(x)-gla) x>(5) f(2) =lim I(r) 5→ag(2)x→ag(x) 负”“返回 下页上页 返回 下页 令f(a)=g(a)=0 于是f(x)及g(x)在点a的某邻域 内连续 在该邻域内应用柯西中值定理，有 (3) ( ) ( ) lim g x f x x a   → 存在(或为无穷大) 那么 ( ) ( ) lim g x f x x→a ( ) ( ) lim g x f x x a   = →  简要证明 ( ) ( ) lim ( ) ( ) ( ) ( ) lim ( ) ( ) lim   g f g x g a f x f a g x f x x a x a x a   = − − = → → → ( ) ( ) lim    g f a   = → ( ) ( ) lim g x f x x a   = →  ( ) ( ) lim ( ) ( ) ( ) ( ) lim ( ) ( ) lim   g f g x g a f x f a g x f x x a x a x a   = − − = → → → ( ) ( ) lim ( ) ( ) ( ) ( ) lim ( ) ( ) lim   g f g x g a f x f a g x f x x a x a x a   = − − = → → → ( ) ( ) lim    g f a   = → ( ) ( ) lim g x f x x a   = →  返回 定理 如果函数f(x)和g(x)满足如下条件 (1) f(x)和g(x)都是当x→a时的无穷小(或无穷大) (2) f(x)和g(x)在点a的某去心邻域内都可导且g(x)0