MATH 110, SECTION 003, WRITTEN ASSIGNMENT 2
Hand in full solutions to the questions below. Make sure to justify all your work, and include complete arguments and
explanations. Your answers must be legible (no tiny drawings or micro-handwriting!). Your answers must be stapled, with
your name and student number at the top of each page. Questions marked with an asterisk are optional. Thinking
about them will improve your understanding of the concepts we have covered, but will not count towards
your grade on the assignment.
(1) Suppose that a is a number such that the limit
x2 + ax + a
x→1
x2 − 1
lim
exists. What is a?
(2) Write down an expression for a function f defined for all real numbers, such that limx→1 f (x) = 4, limx→−1 f (x) = 3,
and f is not continuous at 1.
(3) For each of the following functions, determine whether or not the function is continuous at 1. Explain your reasoning
in full, punctuated sentences.
(a)
f (x) =
x
3
if x 6= 1
if x = 1.
x2
1
.
− 2x + 1
(b)
f (x) =
(c)
ex
ex
if x ≤ 1
if x > 1.
ex
xe
if x ≤ 1
if x > 1.
f (x) =
(d)
f (x) =
(4) Sketch the graph y = x7 − x. Using the theorems discussed in class, prove that the equation
x7 − x = 110
has at least one solution. Write your answer in full, punctuated sentences, being careful to justify each assertion.
(5) Using the theorems discussed in class, prove that if f and g are two functions that are both continuous on [0, 1] such
that f (0) = 0, f (1) = 1, g(0) = 1 and g(1) = 0, then the graphs y = f (x) and y = g(x) must intersect. Write your
answer in full, punctuated sentences, being careful to justify each assertion.
(6∗ ) Suppose that f and g are two functions such that limx→1 (f (x) + g(x)) = 3 and limx→1 (f (x) − g(x)) = 2. Can we
calculate limx→1 f (x)g(x)? Warning: the limits limx→1 f (x) and limx→1 g(x) might not exist.
(7∗ ) Suppose f is a function defined for all real numbers such that for every pair of numbers a and b, and every number
y between f (a) and f (b), there exists a number c between a and b such that f (c) = y. (That is, f satisfies the
conclusion of the intermediate value theorem.) Does it follow that f is continuous?
1