在区间(a, b)上，f(x)和g(x)都可导、g′(x) ≠ 0、limx → a+f(x) = limx → a+g(x) = 0，
$$\lim_{x \rightarrow a^{+}}\frac{f\left( x \right)}{g\left( x \right)} = \lim_{x \rightarrow a^{+}}\frac{f^{'}\left( x \right)}{g^{'}\left( x \right)} $$
证明：设f(a)=g(a)=0，则有limx → a+f(x) = f(a) = 0、limx → a+g(x) = g(a) = 0，所以这个定义使得f(x)和g(x)在[a, b)上连续。取任意的x ∈ (a, b)，由于f(x)和g(x)在[a, x]上满足使用柯西中值定理的条件，所以有
$$\frac{f(x) - f(a)}{g\left( x \right) - g\left( a \right)} = \frac{f'(c)}{g^{'}\left( c \right)}$$
因为f(a)=g(a)=0，所以
$$\frac{f(x)}{g\left( x \right)} = \frac{f'(c)}{g^{'}\left( c \right)}$$
x → a+时，因为c在(a, x)上，所以c → a+，所以
$$\lim_{x \rightarrow a^{+}}\frac{f\left( x \right)}{g\left( x \right)} = \lim_{x \rightarrow a^{+}}\frac{f^{'}\left( c \right)}{g^{'}\left( c \right)} = \lim_{c \rightarrow a^{+}}\frac{f^{'}\left( c \right)}{g^{'}\left( c \right)} = \lim_{x \rightarrow a^{+}}\frac{f^{'}\left( x \right)}{g^{'}\left( x \right)} $$