Winter 2007 - Fitzsimmons' Class - Exam 2

Winter 2007 - Fitzsimmons' Class - Exam 2 - Math 142A(P...

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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 ǫ > 0 there exists δ > 0 such that if x is an element of the domain of f with | x- a | < δ then | f ( x )- L | < ǫ . (b) For each ǫ > 0 there exists δ > 0 such that for all x and y in [ a, b ], if | x- y | < δ then | f ( x )- f ( y ) | < ǫ . 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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Winter 2007 - Fitzsimmons' Class - Exam 2 - Math 142A(P...

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