Student number
Name [SURNAME(S), Givenname(s)]
MATH 100, Section 110 (CSP)
Week 1: Marked Homework Assignment
Due: Thu 2010 Sep 16 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. Find the limit, or else state that the limit does not exist. If you find the limit, show
intermediate steps (explicit justification in terms of limit laws is not required). If you
state that the limit does not exist, give justification.
t2 − 4
t→−2 t + 2
(a) lim
x2 − 3x + 2
x→1 x2 − 2x + 1
(b) lim
(c) lim
x→−1 x2
x−1
− 2x + 1
(d) lim
x→2 x2
x−2
− 4x + 4
1
1
1
−
(e) lim
, where a 6= 0 is some fixed number
h→0 h a + h
a
1
(f) lim
h→0 h
!
1
1
√
− √ , where x > 0 is some fixed number
x
x+h
2. The figure below shows a fixed circle C1 with equation (x − 2)2 + y 2 = 4 and another
circle C2 with variable radius r and centre at the origin. P is the point (0, r), Q is the
upper point of intersection of the two circles, and R is the point of intersection of the
line P Q and the x-axis. What happens to R as C2 shrinks to the origin? Express your
answer as the limit of some suitable function f (r), as r → 0+ , and evaluate the limit.
Hint: Break the problem into several smaller parts, and find a formula for f (r). Then
evaluate the limit.
3. Let
f (x) =



Ax2 + Bx + C if −∞ < x ≤ 0
x3/2 cos(1/x)
if 0 < x < ∞.
where A, B and C are constants. Determine
(a) lim− f (x),
x→0
(b) lim+ f (x),
and include justification. (Note: x3/2 =
x→0
q
(c) f (0),
√
(x3 ) = x x, if x ≥ 0.)