# ASSIGNMENT 1·5 for SECTION 001

```ASSIGNMENT 1&middot;5 for SECTION 001
There are two parts to this assignment. The first part is online 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. The statement
lim f (x) = L
x→a
is defined rigorously as follows:
For any distance ε &gt; 0, there is a distance δ &gt; 0 such that f (x) is within ε of L whenever
x is within δ of a.
Show that
lim 2x − 1 = −3
x→−1
is true according to this definition. In other words, given some ε &gt; 0, explain how you would find a δ &gt; 0
such that 2x − 1 is within ε of −3 whenever x is within δ of −1.
1
, if it exists.
2. Find lim x sin
x→0
x
2
3. Let p and q be functions satisfying
lim p(x) 6= 0 and
x→a
lim q(x) = 0.
x→a
Prove by contradiction that the limit
lim
x→a
does not exist.
p(x)
q(x)
```