October 25, 2005 Lecturer: Dr Martin Kurth Michaelmas Term 2005

```October 25, 2005
Lecturer: Dr Martin Kurth
Michaelmas Term 2005
Course 1E1 2004-2005 (JF Engineers &amp; JF MSISS &amp; 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:
&frac12;
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.
```