This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Prof. Girardi Math 554 Fall 2009 Friday December 11 at 2:00 p.m. Final Exam From our textbook: Introduction to Real Analysis by William F. Trench 2.1.4e Find lim x → x f ( x ) and justify your answer with an- δ proof. f ( x ) = x 3- 1 ( x- 1)( x- 2) + x , x = 1 2.1.7d Find lim x → x- f ( x ) and lim x → x + f ( x ), if they exist. Use- δ proofs, where applicable, to justify your answers. f ( x ) = x 2 + x- 2 √ x + 2 , x = – 2 2.1.11 Prove: If lim x → x f ( x ) = L > 0, then lim x → x p f ( x ) = √ L . 2.1.17c Use an- M definition of lim x →∞ f ( x ) = L to show that lim x →∞ sin x does not exist. 2.1.20b Find, and then prove using the appropriate limit definition, lim x → – 1 x 3 . 2.2.9 The characteristic function ψ T : R → R of a set T is defined by ψ T ( x ) = ( 1 , x ∈ T , x / ∈ T . Show that ψ T is continuous at a point x ∈ R if and only if x ∈ T o ∪ ( T C ) o ....
View Full Document
- Fall '10
- Math, lim g, lim sn, limn→∞ sn