SCIENCE ONE: MATHEMATICS ASSIGNMENT 1
There are two parts to this assignment. The first part is online; you will need to register and login at
www.mathxl.com. The second part consists of the questions on this page. You are expected to provide full
solutions with complete arguments and justifications. You will be graded primarily on the correctness, clarity
and elegance of your solutions. Your answers must be typeset or very neatly written. They must be stapled,
with your name and student number at the top of each page.
1. Let C1 the circle of radius r ≤ 2 centred at the origin, and C2 be the circle of radius 2 centred at (2, 0).
Let P be the point (0, r) on C1 , Q be the upper point of intersection of the two circles, and R be the
point of intersection of the x-axis and the line through P and Q. Explain what happens to R as r → 0.
(You should express your answer as a limit, but you need not use the δ − ε definition of limit to evaluate
it.)
2. Use the δ − ε definition of limit to prove that:
1
1
= , and
(a) lim
x→1 x + 1
2
1
(b) lim sin
does not exist.
x→0
x
(
3. Let f (x) =
1
q
0
if x is rational and of the form
if x is not rational
Find all the points where f is not continuous.
p
q
where the fraction is in lowest terms and q > 0
.