# 7 (chapter 5)(40 points, Due on Friday November 11) Question 1. [3 points each] Use the definition of Limit of functions to prove that x2 2 . 5 5 2 lim x 3 x 1 1 (B) x 1 (A) lim x 0 if x 2 2x (C) lim x 2 f ( x) does not exist where f ( x) 2 if x 2 x Question 2. [3 points each] Use the definition of Uniform Continuity to prove that (A) f ( x) x 3 is uniformly continuous on [0,100] . (B) g ( x) x 2 is not uniformly continuous on R . Question 3. [2 points each] Determine whether the given limits exist and find their values. Give clear explanation. (1) lim x 2 (2) lim x 0 x 4 x 2 |x| x (3) lim x sin( 1 ) x Question 4. [2.5 points each] Determine which of the following functions are uniformly continuous on the given set. Justify your answer. (A) g ( x) 1 on (0,10) . x 2x 1 (B) h( x) on [1,5] . 3x 6 Question 5. [1.5points each] Suppose f , g : D R are two functions. Prove or disprove (1) If D is finite, then f (D ) is finite. (2) If D is bounded, then f is continuous on D . (3) If f 2 is continuous on D , then f is continuous on D . (4) If f g is continuous on D , then f is continuous on D . (5) If f is continuous on D and D is finite, then f is uniformly continuous on D . (6) If D R and f is continuous at each irrational number in D , then f must be continuous at each rational number in D . Question 6. [2.5 points each] Show that (1) 2 x 3 x has a real solution in (0,1). (2) .01x 5 .2 x 4 x 3 .25 0 has a real solution in [-2,2]. Continuous Functions Definition. Let f : D R and c D . We say that f is continuous at c if for each 0 there exits a 0 such that | f ( x) f (c) | whenever | x c | and x D . x sin(1 / x) 0 Example. Show that if f ( x ) x 0 then f is continuous at 0. x 0 Example. Let f : R R be the Dirichlet function defined by 1 f ( x) 0 xQ f is discontinuous at every real number. x Q then Theorem. Let f : D R R and c D . Then the following conditions are equivalent. is continuous at c . (2) f has a limit at c and lim x c f ( x) f (c) (1) f Theorem. Let f , g : D R R and c D . If f , g are continuous at (1) (2) f g, f /g f g are continuous at is continuous at c if c , then c. g (c) 0. Example. Let f , g : D R R be two continuous functions. Then max( f , g ), min( f , g ) are continuous. Follows directly from the above theorem and the 1 2 1 2 1 2 fact that max( f , g ) ( f g ) | f g |, min( f , g ) ( f g ) 1 | f g |. 2 Sequences Definition. A sequence of real numbers is a function f : N R . Examples. 1 1 2 3 2. s n 1 1 / n is a sequence. 1 2 1. x (1, , , ) is a sequence x1 1, x 2 , Definition. A sequence ( s n ) is convergent to s R if for each 0 there exists a real number k such that | s n s | whenever n k . If ( s n ) converges to s n s or s , we write lim n lim s n s Examples. Prove that (1) s n 1 converges to 0. n 2n 2 2n 1 2 . 3 3n 2 5 2 2 2n 2 n 1 Proof of (2). Let s n and s . We need to show that ( s n ) converges to 2 3 3n 5 (2) lim s . Let 0 , take k min{7, If n k , then | 7 }. 9 2 n 2 2n 1 2 6n 2 6n 3 6n 2 10 6n 7 | | || 2 | 2 2 3 3n 5 9n 15 9n 15 6n 7 , but 6n 7 7 n for n 7 and 9n 2 15 9n 2 . So, for n k , we have 2 9n 15 6n 7 7n 7 7 2 2 9 n 9 k 9n 15 9n 7 7 . 9( ) 9 If a sequence does not converge to a real number, it is said to diverge. Examples. Prove that (3) bn ( 1) n is divergent.. (4) lim 2n does not exist. Proof of (4). By contradiction, suppose (bn ) converges to s for some sR. Then for 1 there is k R such that | bn s | 1 for each n k . But for any k R , we can find an odd integer n such that n k . For such n , bn 1 and | bn s | 1 implies that | 1 s | 1 or equivalently 2 s 0 --------------------------------------------------------(1) Similarly, we can find an even integer | bn s | 1 implies that n such that n k . For such n, bn 1 and | 1 s | 1 or equivalently 0 s 2 ------------------------------------------------------------(2) Clearly, (1) and (2) contradict each other. Hence (bn ) is divergent. You are required to read the following pages from the textbook: 161-168 Limit Theorems 1. Every convergent sequence is bounded, the converse is false. 2. If a sequence converges, its limit is unique. 3. (Comparison Test) Let ( s n ), (a n ) be sequences of real numbers and s R. If for some k 0, and some m N , we have | s n s |k | a n | for all n m, and if lim a n 0, then lim s n s. 4. Suppose ( s n ), (t n ) be two convergent sequences such that lim s n s, lim t n t . If s n t n for all n N , then s t. 5. Suppose ( s n ), (t n ) be two convergent sequences such that lim s n s, lim t n t . Then lim( s n t n ) s t lim( s n t n ) st lim(s n / t n ) s / t provided that t n 0, t 0 for all n N . lim ks n ks for ant k R. 6. (Ratio Test Theorem) Suppose ( s n ) be a sequence of positive terms and that the limit L lim exists. If L 1, then lim s n 0. If L 1 then lim s n . 7. Suppose ( s n ) be a sequence of positive terms. Then lim s n iff lim(1 / s n ) 0 s n 1 sn Examples Show that 1 1. lim n n 1 2. If s n is a sequence of nonnegative terms such that lim t n t , then lim t n t n2 3. lim n 0 . 2 n! 4. lim n . 2 n! 5. lim n . n Examples Prove or disprove 1. 2. 3. 4. 5. 6. Every bounded sequence is convergent. If ( s n ) is convergent then (| s n |) is convergent. If (| s n |) is convergent then ( s n ) is convergent. If ( s n ) and (t n ) are convergent sequences then ( s n t n ) is convergent. If ( s n ) and (t n ) are divergent sequences then ( s n t n ) is divergent. If ( s n ) is convergent sequence of rational numbers then lim s n is a rational number 7. If ( s n ) is convergent sequence and lim s n 0 , then s n 10 for each n N . Limits of Functions Definition. lim x c f ( x) L if for each 0 there exits a 0 such that | f ( x) c | whenever 0 | x c | . Examples (1) (2) (3) (4) lim x 2 3 x 6. lim x 2 x 2 4 lim x c x c where c 0. 2 lim x 2 x 2 x 7 7 2x (5) lim x 3 f ( x) does not exist where f ( x) 2 x 1 0 (6) lim x 1 g ( x ) does not exist where g ( x) if x 3 if x 3 if x is rational if x is irrational Limits Theorems. 1. Suppose f , g be two real valued functions such that lim x c f ( x) L, lim x c g ( x) M . Then lim x c ( f g )( x) L M lim x c ( fg )( x) L M lim x c ( f / g )( x) L / M provided that g ( x) 0, M 0 for all x R. lim x c (kf )( x) kL for any k R. 2. Suppose f be a real valued function. Then lim x c f ( x) L iff for every real valued sequence s n that converges to c with s n c for all n , the sequence f ( s n ) converges to L . Examples. Show that x 2 2x 5 (1) lim x 1 2 2 x 3x 5 1 x (2) lim x 0 sin( ) does not exist. Pinching Theorem. Suppose that h( x) f ( x) g ( x) for 0 | x c | where 0 . If lim x c h( x) L lim x c g ( x) then lim x c f ( x) L Examples sin x 1 x 1 cos x 0 (2) lim x 0 x sin 4 x 4 (3) lim x 0 3x 3 (1) lim x 0 Monotone Convergence Theorem A sequence ( s n ) of real numbers is increasing if s n s n 1 , n N . A sequence ( s n ) of real numbers is decreasing if s n s n 1 , n N . A sequence ( s n ) of real numbers is monotone if it is either increasing or decreasing. Monotone Convergence Theorem A monotone sequence is convergent if and only if it is bounded. Examples (1) Let ( s n ) be the sequence defined by s1 1, and s n 1 s n 1 for n 1. Then 1 5 lim s n . 2 Cauchy Sequences Definition. A sequence ( s n ) is a Cauchy sequence if for each 0 there exists a real number k such that | s n s m | whenever n, m k . Theorem. A sequence of real number is a Cauchy sequence iff it is convergent. 1 2 1 3 Example. Show that the sequence s n 1 1 is divergent by showing that it is n not a Cauchy sequence. (2) Let ( s n ) be the sequence defined by s1 3, and s n 1 10 s n 17 for n 1. Show that ( s n ) is convergent and find its limit. Uniform Continuity Definition. Let f : D R R be a function of real numbers. Then f is uniformly continuous on D if for each 0 there exits a 0 such that | f ( x) f (c) | whenever c D and | x c | . Examples (1) f ( x) 3x is uniformly continuous on R (2) f ( x) x 2 is not uniformly continuous on R (3) f ( x) x 2 is uniformly continuous on [0,2] Theorem. If f is continuous on D and D is compact then f uniformly continuous on D . Theorem. Let f : D R R be a function of real numbers. If there are two sequences 1 s n and t n in D such that | s n t n | and | f (sn ) f (t n ) | for some 0 then f is n not uniformly continuous on D . You are required to read the following pages from the textbook:222-226