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October 25, 2005 Lecturer: Dr Martin Kurth Michaelmas Term 2005 Course 1E1 2004-2005 (JF Engineers & JF MSISS & JF MEMS) Problem Sheet 3 Due: in the Tutorials 04 November / 07 November . . . he seemed to approach the grave as a hyperbolic curve approaches a line, less directly as he got nearer, till it was doubtful if he would ever reach it at all. from: Thomas Hardy, Far from the Madding Crowd 1. Let f be a function defined on (−∞, ∞) except at x0 . Write down the formal definition of lim f (x). x→x0 (5 points). 2. Consider the following functions: f (x) = x2 − 9 , x−3 f (x) = ln(x − 3), f (x) = 1 , (x − 3)2 f (x) = x−3 . |x − 3| Calculate lim f (x). x→3 (5 points) 3. A real number x is called rational, if there are two integer numbers m and n such that m x= . n Now consider the following function: ½ 1 if x rational f (x) = 0 otherwise. What is lim f (x)? x→0 (*) Questions 1 and 2 should be answered by all students, you will get points for them. Question 3 is more challenging and meant as an exercise for the more mathematically interested students.