1549901369-Elements_of_Real_Analysis__Denlinger_

4.4 *Infinity in Limits 211 Example 4.4.4 below shows the power of using Theorem 4.4.3 in proving that lim f(x) = +oo. X-+XQ 3x- ...

212 Chapter 4 • Limits of Functions x-2 Example 4.4. 7 Investigate lim --. x-+l X - 1 x-2 Solution: In Example 4.4.5, we showed ...

4.4 *Infinity in Limits 213 (e) Suppose lim f(x) = +oo and lim h(x) = -oo. Let M > 0. Since x-~ x-~ lim f(x) = +oo, 3 81 > ...

214 Chapter 4 • Limits of Functions Similarly, we can combine finite and infinite limits algebraically. Suppose P > 0 and N & ...

4.4 *Infinity in Limits 215 Proof. (a) Exercise 11. (b) Exercise 12. • ALWAYS REMEMBER that +oo and -oo are not real numbers. We ...

216 Chapter 4 • Limits of Functions Prove Theorem 4.4.8 (d). Prove Theorem 4.4.10 (a). Prove Theorem 4.4.10 (b). Show by exampl ...

4.4 *Infinity in Limits 217 . p(x). (a) Prove that if R(x) = q(x), where p(x) and q(x) are polynomials, then the graph of R(x) h ...

218 Chapter 4 • Limits of Functions Theorem 4.4.18 (Fundamental Limits) (a) \:In EN, lim xn = +oo; x-.+oo (b) \:In EN, ~f n is e ...

4.4 *Infinity in Limits 219 Proof. (a) Part 1 (=>):Suppose lim j(x) = L. Let c > 0. Then 38 > 0 3 X-+O+ 1 0 < x < ...

220 Chapter 4 • Limits of Functions (d) Suppose f(x) < 0 for all x in some neighborhood of -oo. Then 1 x~-~ lim f(x ) = -oo & ...

4.4 *Infinity in Limits 221 Theorem 4.4.24 (Limits of Polynomials at ±oo) Let p(x) = anxn + an_ 1 xn-l + · · · + a 1 x + ao deno ...

222 Chapter 4 • Limits of Functions Theorem 4.4.26 (Horizontal Asymptotes of Rational Functions) Con- sider the rational functio ...

4.4 *Infinity in Limits 223 Complete the proof of Example 4.4.22. State and prove a sequential criterion for lim f ( x) = L ...

224 Chapter 4 • Limits of Functions (Project) The Algebra of Limits at Infinity: Prove Theorem 4.4.23. Prove Theorem 4.4.24. 1 ...

Chapter 5 Continuous Functions Sections 5.1- 5.3 are among the most important in the entire book. The ideas discussed here, espe ...

226 Chapter 5 • Continuous Functions 5.1 Continuity of a Function at a Point Definition 5.1.1 (Continuous Function at a Point) S ...

5.1 Continuity of a Function at a Point 227 If Ix - 21 < 1, then -1 < x - 2 < 1, so 1 < x < 3, and so 3 < 3x & ...

228 Chapter 5 • Continuous Functions Proof. Consider the sequence { ~}. Observe that ~ -t 0 and sgn ( ~) 1 -t 1 -=f. sgn(O). Tha ...

5.1 Continuity of a Function at a P oint 229 Proof. Let xo E R For contradiction, suppose f is continuous at x 0. Since the rati ...

230 Chapter 5 • Continuous Functions Proof. (a) Suppose xis an irrational number. Then T(x) = 0. Let n EN. Since x is not ration ...