Final Review, Fall 2002
(1) Find
(a) an explicit irrational number between ln(195) and ln(196).
(b) an explicit rational number between the same two numbers.
(2)
(3)
(a)
(b)
(c)
√
Prove that 2 is irrational.
Suppose that x and y are rational and that Z is irrational. Prove that
x2 + y 2 is rational.
If x 6= 0, then xZ + y 6= 0.
If x 6= 0, then 1/(xZ + y) is irrational.
(4) Let a, b, and c be integers between 0 and 9. Prove that
(a) x = .ab is rational. Hint: Compute x.
(b) y = .cab is rational.
(5)
(a)
(b)
(c)
(d)
State explicit one-to-one correspondences between N and the following sets
The set of rational numbers in the interval (0, 1).
The set of rational numbers in the interval [0, 1].
The set of positive rational numbers.
The set of all rational numbers, including negatives and 0.
(6) Prove that (0, 1) is uncountable.
(7) Define limx→a f (x) = L.
(8) Find the following limits and use δ- to prove your answer.
(a)
(b)
x+1
x→2 3x − 1
lim x2 − 5x + 2
lim
x→1
lim √
(c)
x→2
x
2x + 2
(9) Suppose that limx→a f (x) = 4. Use δ- to prove
p
(a)
lim f (x) = 2
x→a
lim f (x)3 = 43
(b)
(c)
x→a
lim
x→a
f (x)
2
=
f (x) + 2
3
(10) Suppose that limx→a f (x) = L where L > 0. Use δ- to prove
1
1
=
x→a f (x)
L
lim
(11) Suppose that limx→a f (x) = L where L > 2. Prove that there is a δ > 0
such that f (x) > 2 for 0 < |x − a| < δ.
1
2
Remark: It follows that if f (x) ≤ 2 and L = limx→a f (x) exists, then
L ≤ 2.
(12) Find the sup and inf of the following sets and prove your answers.
(a)
(b)
3n
| n ∈ N}
n2 + n + 5
3n
T ={ 2
| n ∈ N}
n +n+5
S = {√
(13) Suppose that for all n ∈ N, an ≥ L and limn→∞ an = L. Prove that
L = inf{an | n ∈ N}.
(14) How should the following function be defined at x = 2 to make it continuous.
√
f (x) =
x+2−2
x2 − 4
(15) Suppose that
f (x) = x
x∈Q
=0
x∈
/Q
(a) Is f continuous at x = π? Explain.
(b) Is f continuous at x = 0? Explain.
(c) a Is f continuous at x = 1? Explain.
(16) Suppose that f (x) is continuous on a closed interval [a, b]. Prove that there
is an M such that f (x) ≤ M for all x ∈ [a, b].
(17) Suppose that f (x) continuous over the interval [a, b] and f (a) < 3 while
f (b) > 3. Let S = {x | f (x) > 3, x ∈ [a, b]}. Let xo = inf S. Prove that
f (xo ) = 3. Hint: This is part of the proof of the IVT.
(18) Prove that there is an x ∈ [0, π/2] such that cos x = x2 .
(19) Suppose that f (x) continuous over the interval [0, 1] and 0 ≤ f (x) ≤ 1.
Prove that there is an x ∈ [0, 1] such that f (x) = x2 .