ASSIGNMENT 1
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. 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 on the correctness, clarity and elegance of
your solutions. Your answers must be typed or very neatly written. They must be stapled, with your name
and student number at the top of each page.
This assignment is due at 10:00 a.m. on Wednesday, September 25. The written assignment is to be
handed in by that time. The online assignment will also close at that time.
1. (a) Prove, using the δ − ε definition of limit, that lim
x→4
√
x = 2.
(b) Let c be a constant and lim f (x) = L. Prove, using the δ − ε definition of limit, that lim cf (x) = cL.
x→a
x→a
2. Let
(
f (x) =
sin
0
1
x
if x 6= 0
if x = 0
.
(a) Prove, using the δ − ε definition of limit, that f is discontinuous at 0.
(b) Consider any interval [a, b] where a ≤ 0 and b > 0, and pick any number L between f (a) and f (b).
Explain why there exists some c in the interval (a, b) such that f (c) = L. Why does this not contradict
the Intermediate Value Theorem?
3. The following questions refer to Figure 3 in the 1999 paper “Hyperaggregation of fish and fisheries: how
catch-per-unit-effort increased as the northern cod (Gadus morhua) declined” from the Canadian Journal
of Fisheries and Aquatic Sciences.
(a) All three functions illustrated in Figure 3 are continuous on their domain. Is this a reasonable
assumption? Defend your answer in one or two paragraphs. Your answer may depend on assumptions
about the physical system being described; if so, you should defend those assumptions.
(b) All three functions illustrated in Figure 3 are also differentiable on their domain. Is this a reasonable
assumption? Defend your answer as in part (a).