1549901369-Elements_of_Real_Analysis__Denlinger_

4.2 Algebra of Limits of Functions 191 Lemma 4.2.10 (Fundamental Limit) For every x 0 E IR., lim x = x 0. Proof. Exercise 3. • T ...

192 Chapter 4 11 Limits of Functions Leto= min{o 1 ,o 2 }. Then, Vx E D(f + g), 0 < Ix - xol < o =? 0 < Ix - xol < 8 ...

4.2 Algebra of Limits of Functions 193 Therefore, lim (f(x)g(x)) = LM. X-+Xo (e) Let E. > 0. Since lim g(x) = M "I-0, g is bo ...

194 Chapter 4 • Limits of Functions Proof. (Alternate) Suppose {xn} is a sequence in [D(f)nD(g)]-{xo} 3 Xn ---; x 0. Since lim f ...

4.2 Algebra of Limits of Functions 195 Finally, we apply the algebra of limits again to conclude that n lim p(x) = lim 2:: akxk ...

196 Chapter 4 • Limits of Functions Proof. Exercise 14. • The following example shows how Theorem 4.2.18 is used in practice. .. ...

4.2 Algebra of Limits of Functions 197 (b) Exercise 16. • In Example 4.1.12 we proved that x -+O lim sin (l) x does not exist. I ...

198 Chapter 4 • Limits of Functions Theorem 4.2.22 (Limits Preserve Inequalities) (a) If lim f(x) = L and f(x) ::::: K for all x ...

4.2 Algebra of Limits of Functions 199 *CHANGE OF VARIABLES IN LIMITS "Change of variables" is a technique you have used frequen ...

200 Chapter 4 • Limits of Functions Theorem 4.2.23 (Change of Variables in Limits) Suppose lim g(x) = x-+xo· uo and lim f(u) = L ...

4.2 Algebra of Limits of Functions 201 EXERCISE SET 4.2 Prove Theorem 4.2.1. Prove Theorem 4.2.5. Prove Lemma 4.2.10. Prove Cas ...

202 Chapter 4 11 Limits of Functions (b) lim [f(x)g(x)] exists, but lim f(x) and lim g(x) do not. X-+Xo X-l-XO X-+Xo (c) lim [f( ...

4.3 One-Sided Limits 203 4.3 One-Sided Limits As you recall from calculus, "one-sided" limits frequently make sense in situa- ti ...

204 Chapter 4 11 Limits of Functions lx-21 Example 4.3.2 Prove that lim -- 2 = -1. x_,2- X - y -l+e Given e -1-e ':_1 lx-2 1 y=- ...

4.3 One-Sided Limits 205 As with Definition 4.3.i, the universal quantifier on x is understood to be present, even when left out ...

206 Chapter 4 11 Limits of Functions Theorem 4.3.5 (Sequential Criterion for One-Sided Limits of Func- tions) (a) lim f(x) = L { ...

4.3 One-Sided Limits 207 Proof. Suppose x 0 is a cluster point of V(f) n(-oo, x 0 ), and a cluster point of'D(J) n(xo,oo). Part ...

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- ...

4.4 *Infinity in Limits 209 13. Revise Theorem 4.2.20 to a correct theorem about limits from the left; limits from the right. 14 ...

210 Chapter 4 a Limits of Functions 1 (b) For arbitrary M > 0, find 8 > 0 3 0 < Ix - 21 < 8 => (x _ 2 ) 2 > M. ...