1549901369-Elements_of_Real_Analysis__Denlinger_

6.6 *L'Hopital's Rule 349

Proof. First note that this theorem covers 30 cases: a= xt, x 0 , x 0 , +oo,

or -oo; L =a real number, +oo, or -oo; and lim g(x) = +oo or -oo.

X->a

Case 1: a= xt, L = a finite real number, and lim g(x) = +oo.

x--+xci

Suppose f, g : I ---+ IR, where f, g, and I satisfy conditions (b )-( e) specified

above.

Let c > 0.

By hypothesis (e), :3 8 > 0 3 xo + 8 EI and

I

f'(x) I

xo < x < xo + 8 * g'(x) - L < c. (13)

Suppose x is any number satisfying x 0 < x < x 0 + 8, and let y be any

number between xo and x. As in the proof of Theorem 6.6.4, the Cauchy mean

value theorem applies to f and g on the closed interval [y,x], so :3cx,y E (y , x)

such that

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

g(x) - g(y) - g'(cx,y) ·

y

I

x

I

Figure 6.11

Thus, by (13) and (14), whenever Xo < x < xo + 8,

I

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

g(x) - g(y) '

which can be transformed algebraically into

f (y) f (x)

-----

g(y) g(y)

1 - g(x)

g(y)

< L+i::.

(14)

(15)

Since this is true for ally between xo and x, we may consider what happens

when y---+ xt. Since g(y)---+ +oo as y---+ xt, we will have 1 - g(x) > 0 as
g(y)
y---+ xt. So inequality (15) is equivalent to

(L - c) [1 - g(x)] + f(x) < f(y) < (L + c) [1 - g(x)] + f(x). (16)

g(y) g(y) g(y) g(y) g(y)