Math 414: Analysis I
Homework 8
Due: April 4, 2014
Name:
The following problems are for additional practice and are not to be turned in: (All
problems come from Basic Analysis, Lebl )
Exercises: 3.1.1-3.1.5, 3.1.8 , 3.1.10
Turn in the following problems.
1. Let f : R → R and let c ∈ R. Show that limf (x) = L if and only if lim f (x + c) = L.
x→c
x→0
2. Use the − δ definition to prove the existence of the following limits.
1
= −1.
x→2 1 − x
(b) lim x3 = c3 .
(a) lim
x→c
3. Show that the following limit does not exist.
lim (x + sgn(x)).
x→0
Here the signum or sign function sgn(x) is defined to be

:x>0
 1
0
:x=0
sgn(x) =

−1 : x < 0
4. Suppose that the function f : R → R has the limit L at 0, and let a > 0. If
g : R → R is defined by g(x) := f (ax) for x ∈ R, show that lim g(x) = L.
x→0
5. Let c ∈ R and let f : R → R be such that lim (f (x))2 = L.
x→c
(a) Show that if L = 0, then lim f (x) = 0.
x→c
(b) Show by example that if L 6= 0, then f may not have a limit at c.
6. Let f : R → R be defined by setting f (x) := x if x is rational, and f (x) = 0 if x is
irrational.
(a) Show that f has a limit at x = 0.
(b) Use a sequential argument to show that if c 6= 0, then f does not have a limit
at c.
7. Let f, g be defined on A ⊂ R → R, and let c be a cluster point of A, Suppose that f
is bounded on a neighborhood of c (i.e. there is a M > 0 such that |f (x)| ≤ M for all
x ∈ (c − δ, c + δ) for some δ > 0), and that lim g(x) = 0. Prove that lim(f g)(x) = 0.
x→c
x→c
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Math 414: Analysis I
Homework 8
Due: April 4, 2014
8. Let f, g be defined on A to R and let c be a cluster point of A.
(a) Show that if both lim f (x) and lim(f (x) + g(x)) exist, then lim g(x) exists.
x→c
x→c
x→c
(b) If lim f (x) and lim(f g)(x) exist, does it follow that lim g(x) exist?
x→c
x→c
x→c
9. Let f : R → R be such that f (x + y) = f (x) + f (y) for all x, y ∈ R. Assume that
lim f (x) = L exists. Prove that L = 0, and then prove that f has a limit at every
x→0
point c ∈ R. Hint: First note that f (2x) = f (x) + f (x) = 2f (x) for x ∈ R. Also
note that f (x) = f (x − c) + f (c) for x, c in R.
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