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Unformatted text preview: Math 142A (P. Fitzsimmons) Second Midterm Exam Solutions 1. Give a definition of each of the following items: (a) lim x a f ( x ) = L (b) f : [ a, b ] R is a uniformly continuous function Solution. (a) For each > 0 there exists > 0 such that if x is an element of the domain of f with | x- a | < then | f ( x )- L | < . (b) For each > 0 there exists > 0 such that for all x and y in [ a, b ], if | x- y | < then | f ( x )- f ( y ) | < . 2. Compute the indicated limits. (Justify your answers.) (a) lim x + x 1 + x 4 (b) lim x + x 2 cos(1 /x ) Solution. (a) Dividing numerator and denominator by x 2 = x 4 we obtain lim x + x 1 + x 4 = lim x + x- 1 x- 4 + 1 = 0 + 1 = 0 . We have used the fact that lim x x- p = 0 if p > 0, as well as several of the rules (sum, root, ratio) for combining limits. (b) In view of the sandwich- x 2 x 2 cos(1 /x ) x 2 , x R , and the known limit lim x 0+ x 2 = 0, the limit in question is equal to 0 because of the...
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