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Unformatted text preview: Time allowed: 20 minutes. 1. (4 marks) The function f is defined by f ( x ) = x 3 for 1 < x 1 and f ( x + 2) = f ( x ) + 2 for all x R . (a) Find an expression for f ( x ) for 3 < x 1. (b) Prove that f is continuous at 1. That is, prove lim x 1 f ( x ) = f ( 1). (c) Does f ( 1) exist? Prove your answer. If it does exist, what is its value? 2. (1 mark) Explain why tan( x + h ) = h + tan( x ) + h tan 2 ( x ) + o ( h ) as h 0. 3. (3 marks) Let f ( x ) = e 2 x . (a) Find lim x f ( x ). (b) Prove your answer in (i) by arguing in terms of and M . 4. (2 marks) State the value of lim x tan x x . Hence, or otherwise, find lim x tan x x x 3 . Week 7 Monday 121 Class 1. f ( x ) = x 3 for 1 < x 1, and f ( x + 2) = f ( x ) + 2 for all x R . (a) For 3 < x 1 f ( x ) = f ( x + 2) 2 = ( x + 2) 3 2 = ( x 3 + 6 x 2 + 12 x + 8) 2 (by the Binomial Theorem) = x 3 + 6 x 2 + 12 x + 6 So for 3 < x...
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This document was uploaded on 03/19/2012.
 Three '11

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