3. limits & continuity - MATH 1804 University Mathematics A Suggested Solutions to Past Exam Problems Chapter 3 Limits and Continuity March 3 2012

# 3. limits & continuity - MATH 1804 University Mathematics A...

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MATH 1804 University Mathematics A Suggested Solutions to Past Exam Problems Chapter 3. Limits and Continuity March 3, 2012 Keywords: · (left/right-hand) limit, sandwich/squeezing theorem · continuity, intermediate value theorem December 2011. 2. (a) For f to be continuous at x = 0, it requires that f (0) = k , lim x 0 + f ( x ) = lim x 0 + x 1 + 3 1 x = 0 = 0 and lim x 0 - f ( x ) = lim x 0 - x 1 + 3 1 x = 0 1 = 0 are equal, i.e. k = 0 . (b) See Chapter 4. Differentiation and Its Applications . May 2011. 1. (a) Note that (as - 1 sin 1 x 1 for x 6 = 0 and sin 2 x 0, we have) - sin 2 x sin 2 x sin 1 x sin 2 x for x 6 = 0 , and lim x 0 ± sin 2 x = 0 , so by the sandwich theorem lim x 0 sin 2 x sin 1 x = 0 . 1
December 2010. 4. (a) Let g ( x ) = x 2 - f ( x ) . Since g ( - 1) = 1 - f ( - 1) = - 3 < 0 and g (2) = 4 - f (2) = 3 > 0 , by the intermediate value theorem , there is some c ( - 1
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