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，
$$lim_{x ightarrow a^{+}}frac{fleft( x ight)}{gleft( x ight)} = lim_{x ightarrow a^{+}}frac{f^{'}left( x ight)}{g^{'}left( x ight)} $$
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
$$frac{f(x) - f(a)}{gleft( x ight) - gleft( a ight)} = frac{f'(c)}{g^{'}left( c ight)}$$
因为f(a)=g(a)=0，所以
$$frac{f(x)}{gleft( x ight)} = frac{f'(c)}{g^{'}left( c ight)}$$
x → a+时，因为c在(a, x)上，所以c → a+，所以
$$lim_{x ightarrow a^{+}}frac{fleft( x ight)}{gleft( x ight)} = lim_{x ightarrow a^{+}}frac{f^{'}left( c ight)}{g^{'}left( c ight)} = lim_{c ightarrow a^{+}}frac{f^{'}left( c ight)}{g^{'}left( c ight)} = lim_{x ightarrow a^{+}}frac{f^{'}left( x ight)}{g^{'}left( x ight)} $$