ASSIGNMENT 3 for SECTION 001
There are three parts to this assignment. Part A is to be completed online before 7:00 a.m. on Friday,
October 8. Part B and Part C, which require full solutions, are to be handed in at the beginning of class on
the same date.
Part A [10 marks]
This part of the assignment can be found online, labelled A3, at webwork.elearning.ubc.ca — sign in using
the MATH110 001 2010W button.
Part B [5 marks]
This part of the assignment is drawn directly from the course texts. It focuses on mathematical exposition;
you will be graded primarily on the clarity and elegance of your solutions.
From the Calculus: Early Transcendentals text, complete question 2 from section 2.2, and question 80 from
section 2.3.
Part C [15 marks]
This part of the assignment consists of more challenging questions. You are expected to provide full solutions
with complete arguments and justifications.
1. The statement
lim f (x) = L
x→a
is defined as follows:
For any distance ε > 0, there is a distance δ > 0 such that f (x) is within ε of L whenever
x is within δ of a.
Show that
lim 2x + 1 = 7
x→3
is true according to this definition. In other words, given any ε > 0, find a δ > 0 such that 2x + 1 is
within ε of 7 whenever x is within δ of 3.
2. A function f is continuous at 0 if lim f (x) = f (0). Determine if the following function is continuous at 0:
x→0
f (x) =
0
1
if x is rational
.
if x is not rational
3. Prove that the following function is continuous at 0:

 0
if x is rational
1
.
f (x) =
 x2 sin
if x is not rational
x