November 1, 2005
Lecturer: Dr Martin Kurth
Michaelmas Term 2005
Course 1E1 2005-2006 (JF Engineers & JF MSISS & JF MEMS)
Problem Sheet 4
Due: in the Tutorials November 11 / November 14
The infinite! No other question has ever moved so profoundly the spirit of man.
David Hilbert (1862-1943)
1. Consider the following function:

 −x3 + 5
0
f (x) =

2x2 − 11
(x < 2)
(x = 2)
(x > 2)
Calculate limx→0− f (x), limx→0+ f (x), limx→2− f (x) and limx→2+ f (x).
Do the limits limx→0 f (x) and limx→2 f (x) exist?
(4 points)
2. Using the Sandwich Theorem, calculate
¶
µ³
cos x ´2
+1 .
lim
x→∞
x
(6 points)
3. Use the Sandwich Theorem to calculate
lim
θ→π/4+
cos2 (θ − π/4).
(*)
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.