Advanced book on Mathematics Olympiad

138 3 Real Analysis

orlimx→x 0 |f(x)|=limx→x 0 |g(x)|=∞, and if additionallylimx→x 0 f

′(x)
g′(x)exists, then
limx→x 0 f(x)g(x)exists and

lim

x→x 0

f(x)

g(x)

=lim

x→x 0

f′(x)

g′(x)

Let us see how L’Hôpital’s rule is applied.

Example.Prove that iff :R→Ris a differentiable function with the property that
limx→∞f(x)exists and is finite, and if limx→∞xf′(x)exists, then this limit is equal
to zero.

Solution.If the limit limx→∞xf′(x)exists, then so does limx→∞(xf (x))′, and the latter
is equal to limx→∞f(x)+limx→∞xf′(x). Applying L’Hôpital’s rule yields

xlim→∞(xf (x))′=xlim→∞

(xf (x))′

x′

=xlim→∞

xf (x)

x

=xlim→∞f(x).

Therefore,

xlim→∞f(x)=xlim→∞f(x)+xlim→∞xf′(x),

and it follows that limx→∞xf′(x)=0, as desired. 

More problems follow.

413.Letfandgben-times continuously differentiable functions in a neighborhood of
a pointa, such thatf(a)=g(a)=α,f′(a)=g′(a),...,f(n−^1 )(a)=g(n−^1 )(a),
andf(n)(a) =g(n)(a). Find, in terms ofα,

xlim→a

ef(x)−eg(x)

f(x)−g(x)

414.For any real numberλ ≥ 1, denote byf (λ)the real solution to the equation
x( 1 +lnx)=λ. Prove that

lim

λ→∞

f (λ)

λ

lnλ

= 1.

3.2.5 The Mean Value Theorem

In the old days, when mathematicians were searching for methods to solve polynomial
equations, an essential tool was Rolle’s theorem.