Calculus: Analytic Geometry and Calculus, with Vectors

132 Functions, limits, derivatives

16 Recall that the signum function having values sgn x (read signum x) is

defined by the formula

sgn x = 1 (x > 0)

sgn x = 0 (x = 0)

sgn x = -1 (x < 0).

Show that

lim sgn x

does not exist. Solution: To prove this without the aid of the result of Problem

15, we let f(x) = sgn x and prove that there is no numberL for which the epsilon-

delta assertion is true. To do this we assume (intending to show that the assump-

tion must be false) that there is a number L for which the assertion is true. Let

e be a number for which 0 < e < 1, and let S be a corresponding positive number

such that l f (x) - LI < e whenever 0 < Ixl < S. If 0 < x < 6, then f(x) = 1

and hence 11 - Ll < e. If - S < x < 0, then f (x) _ -1 and hence -1 - Ll <

e. Therefore,

2=11+11 =11-L+I+LI 511-LI+I1+Ll < 2e

and hence e > 1. This contradicts the inequality e < 1 and establishes our

result.

17 Show that if f(x) = Ixl, then

limf(0 + h) - f(0)

h--0 h

does not exist. Solution: Let g(h) denote the above quotient. When h > 0,

we find that g(h) = h/h = 1, and when h < 0, we find that g(h) -h/h = -1.

The result then follows from the preceding problem.

18 Prove that the first of the assertions

Jim xz = 4, Jim x2 = 5 (?)

x-.2 x- 2

is true and that the second is false.

19 If D is the dizzy dancer function for which

D(x) = 0 (x irrational)

D(x) = 1 (x rational),

prove that there is no a for which lim D(x) exists.

(^20) Suppose that, in some vast universe, it really is true that each flea has a
smaller flea to bite him. Suppose also that the universe contains at least one
flea. Do these hypotheses imply that there exist fleas havingmass less than
1 milligram? ilns.: No. The hypotheses would be satisfied if to each positive
integer n there corresponds a flea whosemass in milligrams is 1 + 1/n, and the
flea of mass I + 1/n is bitten by the flea ofmass 1 +
n I- 1