返回上页 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 ) )( )(l i m xg xf
ax?
存在 ( 或为无穷大 )?
那么 )( )(l i m xg xf
ax? )(
)(l i m
xg
xf
ax?

简要证明
)(
)(lim
)()(
)()(lim
)(
)(lim
g
f
agxg
afxf
xg
xf
axaxax?



)(
)(lim
g
f
a?

)(
)(lim
xg
xf
ax?

)(
)(lim
)()(
)()(lim
)(
)(lim
g
f
agxg
afxf
x
xf
axaxax?


)(
)(lim
)()(
)()(lim
)(
)(lim
g
f
agxg
afxf
xg
xf
axaxax?



)(
)(lim
g
f
a?

)(
)(lim
xg
xf
ax?

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?