8.8. L’HÔPITAL’S RULE 171

Proof: By the definition of limit and 8.20 there exists c < b such that if t > c, then∣∣∣∣ f ′ (t)g′ (t)

−L∣∣∣∣< ε

2.

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 for all s ∈ (a,b) it follows from the 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 letting y→ b−, ∣∣∣∣ f (x)g(x)

−L∣∣∣∣≤ ε

2< ε.

Since ε > 0 is arbitrary, this shows 8.21.The following corollary is proved in the same way.

Corollary 8.8.2 Let [a,b]⊆ [−∞,∞] and suppose f ,g are functions which satisfy,

limx→a+

f (x) = limx→a+

g(x) = 0, (8.22)

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. (8.23)

Then

limx→a+

f (x)g(x)

= L. (8.24)

Here is a simple example which illustrates the use of this rule.

Example 8.8.3 Find limx→05x+sin3x

tan7x .

The conditions of L’Hôpital’s rule are satisfied because the numerator and denomina-tor both converge to 0 and the derivative of the denominator is nonzero for x close to 0.Therefore, if the limit of the quotient of the derivatives exists, it will equal the limit of theoriginal function. Thus,

limx→0

5x+ sin3xtan7x

= limx→0

5+3cos3x7sec2 (7x)

=87.

Sometimes you have to use L’Hôpital’s rule more than once.

Example 8.8.4 Find limx→0sinx−x

x3 .