1549901369-Elements_of_Real_Analysis__Denlinger_

208 Chapter 4 • Limits of Functions

EXERCISE SET 4.3

l. In each of the following, a function f and a number xo are given. Inves-

tigate lim f(x) and lim f(x), and where possible, use the results to

x-x 0 x-xci

determine whether lim f(x) exists.

X--+Xo

(a) f(x) = El; Xo = 0

x

x2

(c) f(x) = ~; xo = 0

Ix+ l l

(b) f(x) = - -; xo = -1

x+l

(d) f(x) = .JX; xo = 0

(e) f(x) = LxJ =the greatest integer^9 :S: x; xo = 3

(f) f(x) = x LxJ; xo = 0 (g) f(x) = x LxJ; xo = 1

2. Revise Theorem 4.1.8 to a correct theorem about limits from the left;
   limits from the right.


3. Revise Corollary 4.1.10 to a correct statement about limits from the left;
   limits from the right.


4. Revise Corollary 4.1.11 to a correct statement about limits from the left;
   limits from the right.


5. Revise Theorem 4.2.1 to a correct theorem about limits from the left;
   limits from the right.


6. Revise Theorem 4.2.5 to a correct theorem about limits from the left;
   limits from the right.


7. Revise Theorem 4.2.9 to a correct theorem about limits from the left;
   limits from the right.


8. Revise Lemma 4.2.10 to a correct statement about limits from the left;
   limits from the right.


9. Revise Theorem 4.2.11 to a correct theorem about limits from the left;
   limits from the right.


10. Revise Theorem 4.2.13 to a correct theorem about limits from the left;
    limits from the right.


11. Revise Theorem 4.2.16 to a correct theorem about limits from the left;
    limits from the right.


12. Revise Theorem 4.2.18 to a correct theorem about limits from the left;
    limits from the right.


13. f is called the "greatest integer function," "bracket function," or "integer floor function."