1549901369-Elements_of_Real_Analysis__Denlinger_

6.6 *L'Hopital's Rule 351 which can be transformed algebraically into f (y) f(x) ----- g(y) g(y) 1 - g(x) g(y) M+l. (19) Since ...

352 Chapter 6 • Differentiable Functions Cases 10, 11, and 12: a = +oo, and L = a (finite) real number, +oo, or -oo, and lim g(x ...

6.6 *L'Hopital's Rule 353 We can use L'Hopital's rule a second time, this time, because we have the indeterminate form 0/0. We g ...

354 Chapter 6 • Differentiable Functions The limits included in Example 6.6.8 were chosen for their straightforward illustrative ...

6.6 *L'Hopital's Rule 355 Use the result of Exercise 2 (b) to prove Cases 13, 14 , and 15 of Theorem 6.6.4. Calculate each of t ...

356 Chapter 6 • Differentiable Functions Prove that \:In E N, lim t: = 0, and use this result to prove that t-tco e . e-1/u -l/ ...

Chapter 7 The Riemann Integral Sections 7.2-7.6 develop the essential core material on the Riemann integral. The Darboux sum app ...

358 Chapter 7 • The Riemann Integral Theorem 7.1.2 Suppose A~ B ~JR, where B is bounded. Then (a) sup A ::::; sup B, and (b) inf ...

7.1 Refresher on Suprema, Infima, and the Forcing Principle 359 Proof. Suppose A, B ~ JR. such that Va E A , Vb E B , a :::; b. ...

360 Chapter 7 • The Riemann Integral Proof. Exercise 5. • EXERCISE SET 7.1 Prove Theorem 7 .1. 2. Prove the remaining part of T ...

7.2 The Riemann Integral Defined 361 a b x (a) (b) Figure 7.1 For i = 1, 2, · · · , n , the ith subint erval of P is [Xi-l, Xi], ...

362 Chapter 7 • The Riemann Integral Theorem 7.2.4 If Q is a refinement of a partition P , then (a) S.(f, Q) 2: S.(f, P), and (b ...

7.2 The Riemann Integral Defined 363 Theorem 7.2.5 If P and Q are any partitions of [a, b], then S__(f, P) ::::; S(J, Q).^2 Proo ...

364 Chapter 7 • The Riemann Integral Finally, we are able to define the definite integral. Definition 7.2.8 ( Darboux's Definiti ...

7.2 The Riemann Integral Defined 365 Proof. Suppose a < b. For every partition P of [a, b], and for every i = 1,2, · · · ,n, ...

366 Chapter 7 • T~e Riemann Integral (b) Let 0 < c; < 1, and let Q = {O, 1-~, 3 + ~, 5}. Then Q is a partition of [O, 5], ...

7.2 The Riemann Integral Defined 367 Proof. (a) Suppose there exists a sequence {Pn} of partitions of [a, b] such that S..(f, Pn ...

368 Chapter 7 • The Riemann Integral 1 Thus, S.(f, Pn) -> 3 as n-> oo. (c) Therefore, by (a), (b), and Theo rem 7.2.12 (c) ...

7.2 The Riemann Integral Defined 369 and n n b S(f,P) = _LMd'~i = ,LJ(xi)~. Hence, i=l i=l n ~ b-a ~ b-a S(f, P) - $._(!, P) = ...

370 Chapter 7 • The Riemann Integral n Then, S(f, P) - S..(f, P) = L(Mi - mi)6i i=l n = L (f(x~') - l(xD) 6.i i=l n < :Lc-6.i ...