MA 301 Test 2, Spring 2006
(1) Dene
lim an = L.
n
6 pts
Solution: No partial credit. It need not be verbatim, but
the meaning must be the same.
Definition 1. Let an be some sequence of numbers and
let L be a number. We say that limn an = L provided that
f
MA 301 Test 3, Spring 2005
TA grades: 1,3,4,5,7
Prof. Grades 2,6,8,9,10
(1) Question 1: Study for Test 4.
State the ocial denition of limxa f (x) = L.
0, 5, 9 or 10 pts. The 9 pts. is if they omit 0 < |x a|.
10 pts
Definition 1. We say that
lim f (x) = L
MA 301 Practice Questions for Test 3, Fall 2006
You should bring a calculator to the test to be able to do
problems such as Problem 12
(1) The following questions refer to Figure 5 on p. 103 of the
notes. Assume that it is given that the graph represents
MA 301 Test 4, Fall 2005
TA Grades 1-4 and 6
(1) State the ocial denition of limxa f (x) = L.
0 or 4 pts.
4 pts
Definition 1. We say that
lim f (x) = L
xa
provided that for all numbers > 0 there is a number > 0
such that
|f (x) L| <
for all x satisfying 0
MA 301 Practice Test 4, Spring 2006
(1) State the ocial denition of limxa f (x) = L.
(2) Suppose that f (x) and g (x) are both continuous at x = a.
Prove that h(x) = f (x)g (x) is also continuous at x = a. You
may use the product theorem for limits of fun
Homework 1 Solutions
You should use the following solutions to grade your work. Give
10 points per problem. Deduct 1 point for each omitted or incorrect
axiom or property. Deduct 1 point if the Assume statement is missing.
Deduct 2 points for not checking
The Field Axioms
(A0) (Existence of Addition) Addition is a well dened process which takes pairs
of real numbers a and b and produces from then one single real number
a + b.
(A1) (Associativity) If a, b, and c are real numbers, then
a + (b + c) = (a + b)
MA 301 Notes: Introduction to Proofs and Real Analysis
Richard C. Penney Purdue University
Contents
Chapter 1. Numbers, proof and `all that jazz'. Chapter 2. Inequalities Chapter 3. Rates of Growth Chapter 4. Limits of Sequences Chapter 5. Limit Theorems
174
CHAPTER 11
Continuity
Consider the problem of measuring the side length of a square and then using the measured date to compute the area of the square. For example, if one side is measured to be 2.74 inches, then the area would be computed as (2.74)2
CHAPTER 1
Numbers, proof and all that jazz.
There is a fundamental dierence between mathematics and other sciences. In most sciences, one does experiments to determine laws. A law will remain a law, only so long as it is not contradicted by experimental e
MA 301 Test 1, Spring 2006
YOU MAY NOT USE THEOREM 2 IN CHAPTER 1
IN ANY OF THE FOLLOWING PROBLEMS.
TA: Grade 1, 5, 6, 7
(1) Dene
8 pts.
lim an = L
n
0, 4 or 8 pts. It need not be verbatim, but the meaning
should be the same.
Solution
Definition 1. We say