1549901369-Elements_of_Real_Analysis__Denlinger_

344 Chapter 6 • Differentiable Functions

Theorem 6.6.3 (L 'Hopital 's Rule) Suppose f, g I --t IR., where I is an

open interval with "endpoint" a, and where

(a) a may be finite, +oo or -oo;

(b) f and g are differentiable on I;

(c) \:/x E J , g(x)g'(x) =J. 0 (that is, neither g(x) nor g'(x) can be 0 on I);

(d) Either (1) J~f(x) = J~g(x) = 0 or (2) IJ~g(x)I = oo;

f'(x)

( e) X-->Q lim -,--( g X ) = L (finite, +oo or - oo).

. f(x). f'(x)
Then X-->Q hm -(-) g X = Xhm -->Q -( gt X -) = L.

Note 1: All the limits shown in the statement of this theorem are one-
sided, since the domains of f and g are restricted to an open interval I with
endpoint a. However, in view of the relationship between limits and one-sided
limits (see Theorem 4.3.8) a proof of this theorem as stated will guarantee that
the same conclusion is true for (two-sided) limits as well.
Note 2: L'H6pital's rule involves the following cases:

• lim f(x) = lim g(x) = 0 or lim g(x) = +oo or -oo;
  x~a x~a x~a
  (3 cases)


• a = xt, x 0 , xo, +oo, or -oo; (5 cases)


• lim f'((x)) = L (finite or +oo, or -oo).
  X-->Q gl X
  (3 cases)

Thus, L'H6pital 's rule will cover 45 different cases, each of which might well
require its own proof! Rather than stating and proving 45 separate theorems,
we shall state only two: one covering 15 cases and a second covering 30 cases.
We shall prove only a small number of these cases. Simple modifications of
these proofs will suffice to prove the remaining cases, and we leave them for
you to prove as exercises.

Theorem 6.6.4 (L'Hopital's Rule I, for 0/0) Suppose f,g: I --t IR., where

I is an open interval with "endpoint" a, and where

(a) a may be finite, +oo or -oo;

(b) f and g are differentiable on I;

(c) \:/x E J, g(x)g'(x) =J. O;