1549901369-Elements_of_Real_Analysis__Denlinger_

6.6 *L'Hopital's Rule 341

(d) Show that if n is odd, Rn-i(x) has opposite signs for x to the left

of a and to the right of a. Show that this implies that f has neither

a local maximum at a nor a local minimum at a.

19. Uniqueness of the Sine and Cosine Functions: Suppose f, g : JR -7 JR
    and s, c : JR -7 JR are functions such that

(a) f' = g and g' = -f;

(c) j(O) = s(O) = O;

(b) s' = c and c' = -s;

( d) g(O) = c(O) = l.

Use Taylor's theorem to prove that f = s and g = c. [Hint: To prove

f = s, apply Taylor's theorem to H(x) = f(x) - s(x) and show^14 that

Blxn+ll

Vn EN, IH(x)I ::; ( )' for some constant B > 0.]

n+ 1.

6.6 *L'Hopital's Rule

This section can be assigned as a project for independent

study.

We are often interested in finding limits of the form

lim j((x)) (8).

X-->Ct g X

Throughout what follows, we allow the limits to be one-sided or even a = +oo
or - oo. In Chapter 4, we saw how to use the "Algebra of Limits" to find the
limit (8) when lim j(x ) and lim g(x) exist but are not both 0 (or oo).
X-+O: X-+O::

INDETERMINATE FORMS 0/0 AND oo/oo

In case lim j(x) = lim g(x) = 0 or lim f(x) = lim g(x) = +oo or -oo,

X-+O: X-+O: X-+O: X-+O:

the limit (8) is called an indeterminate form beca use in these cases lim f((x))
X-->Ct g X
can turn out to be 0, 1, any finite number L, +oo, -oo, or no limit at all, as
the following example shows.

Examples 6.6.1 Observe that all these possibilities can occur, as x -7 o+:

x2

(a) lim - = 0

x-->O+ X

. x
(b) hm - = 1
x-->O+ X

xL

(c) lim - = L

X-->O+ x

x sin l

(f) lim __ x does not exist. D

X-->O+ x

x - x

(d) lim 2 = +oo (e) lim - = -oo

x-->O+ X x-->O+ x^2

14. To see how it is done, look a head to the proof of Theorem 7.7.34.