Bridge to Abstract Mathematics: Mathematical Proof and Structures

4.3 THE LIMIT CONCEPT (OPTIONAL) 143

sin x, x rational

(d) = { 0, x irrational I

ata=O

sin x, x rational

= { 0, x irrational

at a = 42

3. (a) Give an epsilon-delta argument that L f lirn,,, f(x) for each of the fol-
   lowing functions. In each case, indicate the largest value (if any) of e that
   could be used in such a proof:

1, x rational

(ii) f (x) = ~=',a=5

0, x irrational I

*(M) f (x) = 2x-B, xIO L = 0, a = 0, where B >^0

x rational

(jV) = {:: x irrational L=l,a=l

(6) Argue that, for any real number L, L # lirn,,, f(x), where f(x) is the function

given in Example 2. (See especially the last paragraph of the solution to Example

2.)

4. Let f be a function defined on the interval [a, a + r) for some r > 0. A real number
   L+ is said to be a right-hand limit of f(x), as x approaches a, denoted L+ = lim,,,,
   f(x), if and only if (Ve > 0)(36 > O)(Vx)[(a < x < a + 6) -r (I f(x) - L+I < e)]. Left-
   hand limit L- off at a is defined analogously; write out this definition. It is a theo-
   rem, to be proved in Article 6.1 (see Exercise 19), that lirn,,, f(x) exists if and only

/'

if lim,,,+ f(x) and lim,,,- f(x) both exist and are equal. Evaluate both the right-

and left-hand limit as x tends to a for each of the following functions:

5x + B'

(a) = {h - 8, X 1 > O}, 0 a = 0, where B >^0

(b) f (x) = [XI, the greatest integer less than or equal to x, a = 1

*(d) f (x) = tan x, a = n/2

x rational

} a=*

= {: x irrational