ASSIGNMENT 1·6
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 typeset or very neatly written. They must be stapled, with your name
and student number at the top of each page.
1. Come up with an expression for a function f such that the domain of f is all real numbers, lim f (x) = 3,
x→2
and f (2) = −3.
1
6= 0.
x→0
x
1
= 0.
(b) Explain why lim x sin
x→0
x
2. (a) Explain why lim sin
3. Consider a square of side length 1. In each of the three pictures below, the solid red line forms a path
from one corner to the corner diagonally opposite. Call these, from left to right, the 1-step staircase, the
1
1
2 -step staircase, and the 3 -step staircase (so called for the length of each step). Let f (n) be the total
length of the n-step staircase. This is a function with domain n = 1, 21 , 13 , 14 , . . .. Is
√
lim f (n) = 2?
n→0
Justify your answer.