1549901369-Elements_of_Real_Analysis__Denlinger_

186 Chapter 4 • Limits of Functions

Proof. Let f ( x) = sin ( ~). Consider the sequences { Xn} = { nl7f } and

{yn} = {

1

}. Then both Xn ____, 0 and Yn ____, 0, but

~ + 2mr

f(xn) = sin ( x

1

n) = sin(mr) = 0 ____, 0, while

f (Yn) = sin ( y~) = sin(~ + 2mr) = 1 ____, 1.

Thus, lim f(xn) -I- lim f(Yn)· Therefore, by Corollary 4.1.11, lim f(x)
n--+oo n--+CXJ x--+0
does not exist. D

llx

-1/x

y =sin (1/x)

Figure 4.2

EXERCISE SET 4.1

1. For each of the following limit statements lim f ( x) = L, do the following:
   X--+XQ
   (i) Find a value of b > 0 that will guarantee that whenever x is within
   distance b from xo (but -I-xo) f(x) is within distance. 01 from L.
   (ii) Find a value of b > 0 that will guarantee that whenever x is within
   distance b from Xo (but not -I-xo) f(x) will approximate the limit
   accurately to 3 decimal places.
   (iii) For arbitrary but unknown E > 0, find a value of b > 0 that will
   guarantee that whenever x is within distance b of x 0 (but -I-xo)
   f(x) is within distance of E of L.
   (iv) Prove the given limit statement using Definition 4.1.1
   (a) lim(5x - 11) = 4 (b) lim(3x - 8) = - 5
   x-1-3 X--+l
   (c) lim x^2 = 9 (d) lim x^3 = 8
   x--+3 x --+2

x