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MATH 131
Homework 2
(Due date: Thursday, Oct 17)
Please do the following problems. Make sure to show all details of your work.
1. (16 pts) Find all real numbers x such that (4 pts for each part)
(a) x 2 + 3x + 2 = 0.
(b) x 2 + 4x + 2 = 0.
1
(c) x + = 2.
x
1
(d) x + = 0.
x
2. (10 pts) Let a and b be real numbers. Find all real numbers x (in terms of a
and b) such that
(x − a)(x − b) > 0,
when (5 pts for each part)
(a) a < b.
(b) a = b.
3. (24 pts) Draw a graph of f and find lim x→2 f (x) where f : R → R is (6 pts
each; 2 for the graph and 4 for the limit)
(a) f (x) = 2x.
(b) f (x) = |x − 2|.
¨
4x − 4, when x ≤ 2,
(c) f (x) =
4,
when x > 2.
¨
x 2,
when x 6= 2,
p
(d) f (x) =
2 2, when x = 2.
4. (12 pts) Use the rigorous definition of limit to justify your answer for parts
(a) and (b) in the question above (6 pts for each part).
5. (4 pts) Let x, y, z be three real numbers, then the triangle inequality says
that
|x + y| ≤ |x| + | y|.
(1)
What does this statement says geometrically. In particular, why is it called
the triangle inequality?
1
6. (extra credit; 8 pts) Prove the triangle inequality.
(HINT. Divide into different cases and prove for each case separately. Alternatively, you can square both sides of (1). If you choose to do this, please
justify why squaring both sides is allowed.)
7. (4 pts) Let f , g : R → R be functions and x 0 a real number such that the
limits of f and g exist at x 0 . Use the triangle inequality to show that
lim ( f (x) + g(x)) = lim f (x) + lim g(x).
x→x 0
x→x 0
2
x→x 0