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Unformatted text preview: Time allowed: 20 minutes. 1. (4 marks) Let f ( x ) = x 2 . (a) If a > 0, find lim x a f ( x ). (b) Prove your answer in (i) by arguing in terms of and . 2. (2 marks) Find lim x 1 cos 3 x x 2 . 3. (2 marks) Consider the function f ( x ) = x 4 4 x 3 11 x 2 + 30 x x a . This is not continuous at x = a . For what values of a is the discontinuity removable? 4. (2 marks) State the Mean Value Theorem. Hence or otherwise, show that x sin 1 x for 1 < x 0. Version 2, 4/5/08 Week 7 Tuesday 10-11 Class 1. (a) lim x a f ( x ) = a 2 . (b) lim x a f ( x ) = l means: For all real numbers > 0, there exists a real number > 0, such that: | f ( x ) l | < whenever 0 < | x a | < . Let > 0 be given. We want to find > 0 such that | x 2 a 2 | < whenever 0 < | x a | < . Now | x 2 a 2 | = | x a || x + a | . Lets assume | x a | < a....
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- Three '11