SOLUTION: Emirates National School Functions and Graphs Exam Practice

UMC Advanced Calculus Numerical Sequences Inequalities & Number Sets Exam Practice

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).