1549901369-Elements_of_Real_Analysis__Denlinger_

6.6 *L'Hopital's Rule 345

(d) X-+Q lim f(x) = X-+Q lim g(x) = O;

f'(x)

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

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

P roof. First, note that this theorem covers 15 cases: a = xt, x 0 , x 0 , +oo,
or -oo, and L = a (finite) real number, +oo, or -oo.

Case 1: a= xt, and L = a (finite) real number.
Suppose f,g: I_, JR, where f , g, and I satisfy conditions (b)- (e) specified
above.

Let c > 0. Then 3 6 > 0 3 Xo + 6 E I and

I

f'(x) I

x 0 < x < xo + 6 =? g'(x) - L < c. (9)

Suppose x is any number satisfying xo < x < xo + 6, and let y be any
number between x 0 and x. The Cauchy mea n value theorem applies to f and
g on the closed interval [y, x] since f and g are differentiable on the interval I,
which contains x and y. Since g'(x) i=- 0 on I , the Cauchy mean value theorem
guarantees 3cx,y E (y,x ) such that

f(x) - f(y) f'(cx,y)

g(x) - g(y) g' ( Cx,y).

y x

I

+

I I

xo xo+8

Cx,y

Fig ure 6.9

(Note that g' i=- 0 on I , and hence by Rolle's theorem, g(x) - g(y) i=-0.)

Thus, by (9) and (10),

I

f(x) - f(y) - LI < c.

g(x) - g(y)

Since this is true for all y E ( x o, x ), we have

lim I f(x) - f(y) - LI < c

y->xci g(x) - g(y)

. i.e., lf(x)g(x) - 0 - 0 - L I < c.

I

Thus, x f(x) I. f(x )

0 < x < x^0 +^6 =? -g (-) X - L < c. Therefore, X->x;i hm - (- ) g X = L.

(10)