exams - MAT 370/371 Exams Spring 2002 Exam 1 1(15 pts...

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Unformatted text preview: MAT 370/371 Exams Spring 2002 Exam 1 1. (15 pts) Definitions. (a) Give the precise definition of lim x → x f ( x ) = L. (b) Give an example where such a limit does not exist. (c) Let the domain of the function f be N , the natural numbers, and let f ( x ) = x 2 . What can you say about lim x → 3 f ( x )? 2. (10 pts) Define carefully: (a) Accumulation point. (b) Subsequence 3. (15 pts) Use the definition of convergence directly to show that lim n →∞ n n + 1 = 1 . 4. (10 pts) State the Bolzano-Weierstrass theorem. 5. (10 pts) Find the following limit and justify your answer: lim n →∞ 2 n + 3 n + sin( n ) /n = 1 . DO 2 OF THE FOLLOWING 3 PROBLEMS 6. (20 pts) Use mathematical induction to prove that 1 + 3 + 5 + 7 + ··· + (2 n- 1) = n 2 7. (20 pts) Prove ( A ∪ B ) ∩ C = ( A ∩ C ) ∪ ( B ∩ C ) . 8. (20 pts) Let S ⊂ R and let q be the greatest lower bound of S . Suppose that q 6∈ S . Prove that q must be an accumulation point of S Exam 2 In doing proofs, when using a major theorem, quote it as best you can. 1. (15 pts) Definitions. (a) Give the precise definition of continuity of a function f at a point x . (b) Define increasing function . (c) Define lim x → x f ( x ) = L. 1 2. (15 pts) Prove, using an ²- δ argument that lim x → x 1 / 3 [sin( x ) + 2cos( x )] = 0 . 3. Let us define the function g by: g ( x ) = 1 when x is irrational and g ( x ) = 0 if x is rational. Show that lim x → g ( x ) does not exist. 4. (15 pts) Use an ²- δ argument to prove that lim x →- 4 5 x 2 + 25 x + 20 ( x + 4) =- 15 . 5. (15 pts) Prove that if f : D → R is continuous at x and if { x n } ∞ n =1 converges to x then lim n →∞ f ( x n ) = f ( x ) . DO 2 OF THE FOLLOWING 3 PROBLEMS 6. (15 pts) Let f : [ a,b ] → R be an increasing function. Show that lim x → b f ( x ) exists 7. (15 pts) Let f , g and h be three functions from the domain D ⊂ R to R such that f ( x ) ≤ g ( x ) ≤ h ( x ) for all x ∈ D . Suppose that x is an accumulation point of D and that lim x → x f ( x ) = L = lim x → x h ( x ) . Prove that lim x → x g ( x ) = L. 8. (15 pts) Let x be an accumulation point for D and f : D → R a function with the property: Whenever { x n } ∞ n =1 is a sequence of points from D \ { x } that converges to x then the sequence { f ( x n ) } ∞ n =1 converges. Let { u n } ∞ n =1 and { v n } ∞ n =1 be two sequences in D converging to x such that for all n ∈ J we have u n 6 = x 6 = v n . Prove that lim n →∞ f ( u n ) = lim n →∞ f ( v n ) . Exam 3 1. (20 pts) Give the precise definition of the Riemann integral . In the process you will need to define partition, upper and lower Riemann sums and upper and lower Riemann integrals ....
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exams - MAT 370/371 Exams Spring 2002 Exam 1 1(15 pts...

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