1549901369-Elements_of_Real_Analysis__Denlinger_

666 Appendix C • Answers & Hints for Selected Exercises

1 1

(f) lim .;x - vxo = lim Vx - ftO = lim -l

X->Xo X - XQ X->Xo vxftO(X - XQ) X->Xo vxftQ( Vx + ftQ)

-1

2xoft0

()

. J4(x+h)-1-y'4x-l. (4x+4h-1)-(4x-1)
2. e hm = hm r~::;:::::==~===--;:::::::=::,-
h->O h h->Oh[J4(x+h)-l+y'4x-l]
4h 2

• lim - ---===


• h-^0 h [J4(x+h)-l+v'4x-l] - v'4x-1·

3. (b) lim f(x) - f(xo) = lim Q = lim 0 = 0, and
   x->O- x - Xo X->O- x x-o-

lim f(x) - f(xo) = lim x

2

= lim x = O·

x-+O+ X - Xo x-+O+ X x-+O+ '

:. lim f(x) - f(xo) = O.

x-+0 X - XO

( c) yes ( d) no

5. (a) lim I f(x) - f(xo) I = lim I f(x) I ::::; lim xi

2

1

= lim lxl = 0. :. f is differ-
x-+O X - Xo x-+O x x-+O x x-+0
entiable at 0 and f' (0) = 0.

(b) Suppose x -=/= 0. Then 3 sequence {rn} of rational numbers converging
to x, and a sequence {Zn} of irrationals converging to x. If x is rational, then

lim I f(zn) - f(x) I = lim IO -x

2
I = +oo, so f is not differentiable at x. If
n-+oo Zn - X n->oo Zn - X

x lS.. irrat10na. 1 h , t en l" lm I f(rn) - f(x) I = l" lm Ir~ ---^0 I = +oo, so f. lS not
n-+oo rn - x n-+oo rn - x
differentiable at x. :. f is not differentiable at any x -=/= 0.

7. (a) Continuous everywhere; differentiable ¢=? x-=/= 0; f'(x) = 0 if x < 0, 2 if
   x > O; f' continuous ¢=? x -=/= 0.
   (c) Continuous everywhere; differentiable ¢=? x -=/= mr (n E Z); f'(x) =
   (-1r cosx if x E (mr, (n + l)n); f' continuous¢=? xi= nn.
   ( e) Continuous on ( -1, 1) and on ( n, n+ 1), n E Z. \In E Z, f is differentiable
   on (n, n + 1), f' (x) = n on (n, n + 1), and f' is continuous there.


8. (a) lim J(x) = lim xrsin~.
   x->O+ x-+O+
   (i) If r > 0, lim xr = 0 and sin l is bounded, so by Ex. 4.2.19, lim xr sin l =
   x->O+ x x->O+ x
   0 = f(O). Thus, f is continuous from the right at 0.
   (ii) If r = 0, lim f(x) = lim sin~' which does not exist.
   x->O+ x-+O+
   (iii) If r < 0, let Xn = "+ 2 12 n?T. Then Xn --+ o+ but f (xn) = ( ~+2n7r^1 r --+ 00.