5.17. L’HÔPITAL’S RULE 149
Now pick x,y such that c < x < y < b. By the Cauchy mean value theorem, there existst ∈ (x,y) such that
g′ (t)( f (x)− f (y)) = f ′ (t)(g(x)−g(y)) .
Since g′ (s) ̸= 0 on (a,b) , it follows from mean value theorem g(x)−g(y) ̸= 0. Therefore,
f ′ (t)g′ (t)
=f (x)− f (y)g(x)−g(y)
and so, since t > c, ∣∣∣∣ f (x)− f (y)g(x)−g(y)
−L∣∣∣∣< ε
2.
Now this implies ∣∣∣∣∣∣ f (y)g(y)
(f (x)f (y) −1
)(
g(x)g(y) −1
) −L
∣∣∣∣∣∣< ε
2
where for all y large enough, both f (x)f (y) −1 and g(x)
g(y) −1 are not equal to zero. Then
∣∣∣∣∣∣ f (y)g(y)
−L
(g(x)g(y) −1
)(
f (x)f (y) −1
)∣∣∣∣∣∣< ε
2
∣∣∣∣∣∣(
g(x)g(y) −1
)(
f (x)f (y) −1
)∣∣∣∣∣∣ .
Therefore, for y large enough,
∣∣∣∣ f (y)g(y)
−L∣∣∣∣≤∣∣∣∣∣∣L−L
(g(x)g(y) −1
)(
f (x)f (y) −1
)∣∣∣∣∣∣+ ε
2
∣∣∣∣∣∣(
g(x)g(y) −1
)(
f (x)f (y) −1
)∣∣∣∣∣∣< ε
due to the assumption 5.14 which implies limy→b−
(g(x)g(y)−1
)(
f (x)f (y)−1
) = 1. Therefore, whenever y is
large enough,∣∣∣ f (y)
g(y) −L∣∣∣< ε and this is what is meant by 5.16.
As before, there is no essential difference between the proof in the case where x → b−and the proof when x → a+. This observation is stated as the next corollary.
Corollary 5.17.9 Let [a,b]⊆ [−∞,∞] and suppose f ,g are functions which satisfy,
limx→a+
f (x) =±∞ and limx→a+
g(x) =±∞, (5.17)
and f ′ and g′ exist on (a,b) with g′ (x) ̸= 0 on (a,b). Suppose also that
limx→a+
f ′ (x)g′ (x)
= L. (5.18)
Then
limx→a+
f (x)g(x)
= L. (5.19)