1549901369-Elements_of_Real_Analysis__Denlinger_

346 Chapter 6 1111 Differentiable Functions

Case 2: a = xt, and L = +oo.

Again, suppose f ,g: I---->~, where f,g, and I satisfy conditions (b)- (e)

specified above.

Let M > 0. Then 3 6 > 0 such that xo + 6 E I and

f'(x)

x 0 < x < xo + 6 =? g'(x) > M. (11)

Suppose x is any number satisfying xo < x < xo + 6, and let y be any
number between x 0 and x. As in the proof of Case 1, the Cauchy mean value
theorem applies to f and g on the closed interval [y, x] so 3 Cx,y E (y, x) such
that

Thus, by (11) and (12),

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

g(x) - g(y) - g'(cx,y) ·

y

I

x

I

Figure 6.10

xo+o

(12)

f(x) - f(y)

xo < x < xo + 6 =? g ( x ) -g ( ) y > M.

As in the proof of Case 1, we take the limit as y----> xt, and obtain

f (x)

xo < x < xo + 6 =? g(x) ;::: M.

. f(x)
Therefore, hm -( -) = +oo.
x-->x(j g X
Case 3: a= xt, L = -oo. (Exercise 1.)
Cases 4, 5, and 6: a = x 0 , L = a (finite) real number, +oo, or -oo.
(Exercise 3)
Cases 7, 8, and 9: a= xo, L = a (finite) real number, +oo, or -oo.
(Exercise 4)
Cases 10, 11, and 12: a= +oo, and L = a real number, +oo, or -oo.
Our hypotheses assure us that f and g are differentiable over some interval
(a, +oo), with a> 0. We can use Cases 1- 3 to treat Cases 10 - 12 if we make