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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- 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- 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 T : R R of a set T is defined by T ( x ) = ( 1 , x T , x / T . Show that T is continuous at a point x R if and only if x T o ( T C ) o ....
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This note was uploaded on 12/13/2011 for the course MATH 554 taught by Professor Girardi during the Fall '10 term at South Carolina.
- Fall '10