UMC Advanced Calculus Numerical Sequences Inequalities & Number Sets Exam Practice

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In Questions 1-5 prove your answer. In this test you can use
without proofs theorems that were proved in the lectures or in the
book, just give a reference.
1.(10 pts) Let S be a subset of R, and u ∈ R. Two of the following
statements, if combined, imply that u = inf(S). Which two statements?
A: u ≤ s for every s ∈ S.
B: There exists ε > 0 and s ∈ S so that s < u + ε. C: For every ε > 0, the number −u−ε is not an upper bound of the
set−S={x∈R: −x∈S}.
D: There exists ε > 0 so that u + ε is not a lower bound of S.
1
2
and let ε > 0. Suppose that |xn| < M for every n. Select one of the following sets of constants ε1 and ε2 for which the inequalities 2. (10 pts) Let {xn} and {yn} be two sequences, limxn =L1 ̸=0, limyn =L2 ̸=0, |xn −L1|<ε1 imply |xnyn − L1L2| < ε. εε A: ε1 = 2M , ε2 = 2|L1|. εε B: ε1 = 2|L1|, ε2 = 2M . εε C: ε1 = 2|L2|, ε2 = 2M . εε D: ε1 = 2M , ε2 = 2|L2|. and |yn −L2|<ε2 3 3. (10 pts) Which of the following statements imply that a sequence {xn} does not have a limit: A: for every ε > 0 there exist n, m ∈ N such that |xn − xm| > ε.

B: there exist a natural number K and ε > 0 such that for every
n > K we have |xn − xK | > ε.
C: {xn} is an increasing sequence and |xn| > 1000 for every n > 1000.
D: there exists ε > 0 such that for every natural number K there
exist m, n > K with |xm − xn| > ε.
4
4.(10 pts) Which two of the following statements combined imply
that limx→3 f(x) = 2?
A: limn→∞ f (3 + 1 ) = limn→∞ f (3 − 1 ) = 2.
nn
B: limx→3+ f(x) = 2

C: limx→3+ f(x) and limx→3− f(x) exist

D: For every sequence xn → 3 with xn ≥ 3 for every n, one has
lim f(xn) = 2.
n→∞
5
5. (10 pts) Which two of the following statements combined imply
that the equation f (x) = 1 has a solution in the interval [1, 3]?
A: 0 < f(1) and f(3) < 2. B: f is continuous on [1, 3]. C: f(1) < 0 and f(3) > 2.
D: f is monotone on [1, 3].
6
6. (10 pts) Suppose that lim|xn| = 3, but {xn} does not have a
subsequence with limit 3. Prove that lim xn = −3.
7
7. (10 pts). Suppose {xn} and {yn} are bounded sequences and for
every n ∈ N
xn + yn+1 ≤ xn+1 + yn,
and

Prove that both sequences {xn} and {yn} converge.
xn + yn ≥ xn+1 + yn+1.
8
8. (10 pts) Prove using the definition that the following limit is equal
to −∞ :
lim x+1 =−∞.
x→2− x2 − 4
9
9. (10 pts) Prove that if a function f is non-negative and continuous
on the interval [1, ∞), and limx→∞ f (x) = 0, then there exists xM ∈
[1,∞) such that f(xM) ≥ f(x) for every x ∈ [1,∞).
10
10. (10 pts) Prove that the function f(x) = x1/3 is uniformly con-
tinuous on [1, ∞).
Hint: Use x − y = (x1/3 − y1/3)(x2/3 + x1/3y1/3 + y2/3).

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EXAM 1. ADVANCED CALCULUS I, FALL 2020
Name:
In Questions 1-5 prove your answer. In this test you can use
without proofs theorems that were proved in the lectures or in the
book, just give a reference.
1.(10 pts) Let S be a subset of R, and u ∈ R. Two of the following
statements, if combined, imply that u = inf(S). Which two statements?
A: u ≤ s for every s ∈ S.
B: There exists > 0 and s ∈ S so that s < u + . C: For every > 0, the number −u − is not an upper bound of the
set −S = {x ∈ R : −x ∈ S}.
D: There exists > 0 so that u + is not a lower bound of S.
1
2
2. (10 pts) Let {xn } and {yn } be two sequences,
lim xn = L1 6= 0,
lim yn = L2 6= 0,
and let > 0. Suppose that |xn | < M for every n. Select one of the following sets of constants 1 and 2 for which the inequalities |xn − L1 | < 1 imply |xn yn − L1 L2 | < . A: 1 = B: 1 = C: 1 = D: 1 = , 2 = 2|L 1 | . 2M , 2 = 2M . 2|L1 | , 2 = 2M . 2|L2 | , 2 = 2|L 2 | . 2M and |yn − L2 | < 2 3 3. (10 pts) Which of the following statements imply that a sequence {xn } does not have a limit: A: for every > 0 there exist n, m ∈ N such that |xn − xm | > .
B: there exist a natural number K and > 0 such that for every
n > K we have |xn − xK | > .
C: {xn } is an increasing sequence and |xn | > 1000 for every n > 1000.
D: there exists > 0 such that for every natural number K there
exist m, n > K with |xm − xn | > .
4
4.(10 pts) Which two of the following statements combined imply
that limx→3 f (x) = 2?
A: limn→∞ f (3 + n1 ) = limn→∞ f (3 − n1 ) = 2.
B: limx→3+ f (x) = 2
C: limx→3+ f (x) and limx→3− f (x) exist
D: For every sequence xn → 3 with xn ≥ 3 for every n, one has
lim f (xn ) = 2.
n→∞
5
5. (10 pts) Which two of the following statements combined imply
that the equation f (x) = 1 has a solution in the interval [1, 3]?
A: 0 < f (1) and f (3) < 2. B: f is continuous on [1, 3]. C: f (1) < 0 and f (3) > 2.
D: f is monotone on [1, 3].
6
6. (10 pts) Suppose that lim |xn | = 3, but {xn } does not have a
subsequence with limit 3. Prove that lim xn = −3.
7
7. (10 pts). Suppose {xn } and {yn } are bounded sequences and for
every n ∈ N
xn + yn+1 ≤ xn+1 + yn ,
and
xn + yn ≥ xn+1 + yn+1 .
Prove that both sequences {xn } and {yn } converge.
8
8. (10 pts) Prove using the definition that the following limit is equal
to −∞ :
x+1
= −∞.
x→2− x2 − 4
lim
9
9. (10 pts) Prove that if a function f is non-negative and continuous
on the interval [1, ∞), and limx→∞ f (x) = 0, then there exists xM ∈
[1, ∞) such that f (xM ) ≥ f (x) for every x ∈ [1, ∞).
10
10. (10 pts) Prove that the function f (x) = x1/3 is uniformly continuous on [1, ∞).
Hint: Use x − y = (x1/3 − y 1/3 )(x2/3 + x1/3 y 1/3 + y 2/3 ).

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Tags:
Advanced calculus

numerical sequences

Inequalities

Number Sets

proofs theorems

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