Math 214
Homework for Sections 1.5 and 1.6
Spring 2016
This problem set is out of 45 points. It is due on Thursday, March 3.
Problem 1. (2 points each) Give an example of each of the following types of subsets of
Rn or argue that it cannot exist:
1. A set that is both open and closed.
2. A set that is neither open nor closed.
3. A closed set that is not compact.
4. A bounded set that is not compact.
5. A closed set with empty interior.
Problem 2. (5 points each) For each of the following limits, use a theorem or a definition
to justify whether it exists or not. If it exists, evaluate it.
1. lim(x,y,z)→(1,3,2)
2. lim(x,y)→(0,0)
x2 +y 3 −xyz
.
ex+y−2z
x2 +2y 2
.
x2 +y 2
3. lim(x,y)→(0,0) √ xy
2
x +y 2
.
Problem 3. (10 points) Prove that if h : Rn → R is a function which is continuous at
x0 ∈ Rn and h(x0 ) 6= 0, then the function whose rule is h1 is also continuous at x0 .
Problem 4. (10 points) Complete Problem # 1.6.6 from the book.
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