Answer:A: fEy0y4

Question: Let f and g be the functions whose values at xR are given by fx2x and gx3x2 1The composite function gcirc f is

Answer:A: gcirc fx12x2 1

Question: Determine the set A of all real numbers x such that 2x36

Answer:A: x32

Question: If In0 frac1n for nin mattabN then In subseteq In1 for each nnin mattabN so that this sequence of intervals is

Answer:A: Nested

Question: A positive real number is rational if and only if its decimal representation is

Question: The following statements about an upper bound U of S are equivalent except

Answer:A: Periodic

Question: The unit interval 0 1 xR0x1 is

Answer:A: Not countable

Question: If C is an infinite set and B is a finite set then CB is an

Answer:A: infinite set

Answer:A: If tU then t is not an upper bound of S

Answer:A: frac1n2 n1leq frac1n2

Question: If xn is an unbounded increasing sequence then limxn is

Answer:A: infty

Question: Consider the sequence by 1nn0infty sumn0ifty 1n1 11

Answer:A: Divergent

Question: The series sumn1infty frac1n2 n1 is convergent if the inequality

Question: The compliment of B relative to A is the set

Question: If A123 andB45 then the set AB is the set whose elements are the ordered pairs

Answer:A: ABxxA and xB

Question: If A B and C are sets then ABC is equal to

Answer:A: ABAC

Question: If A and B are non empty sets then the Cartesian product of AB of A and B is the set of all ordered pair ab with aA and bBThat is

Answer:A: ABabaAbB

Answer:A: 14 15 24 25 3435

Question: A nonempty subset p of is called the set of positive real numbers the following are the order properties of except

Question: A function meant a definite formula such as fxx23x50 which associates to each real number x another number fx then find f 4

Answer:A: 9

Question: A function which is both one one and onto is known as

Answer:A: Bijective

Question: In the distributive law A BC is equal to

Answer:A: AB AC

Answer:A: If abPab1

Question: Let a b c be any elements of the following are equivalent except

Question: Consider the series sumn1inftyfrac1nn1frac112frac123frac134 this series converges to

Answer:A: If ab and c0 then cacb

Question: The limit of the sequence frac1n21 is

Answer:A: 1

Question: The limit of the sequence frac2nn2 is

Answer:A: 2

Question: The Bolzano Weierstrass theorem say that a bounded sequence of real numbers has a

Answer:A: Convergent subsequence

Question: A monotone sequence of real number is properly divergent if and only if it is

Answer:A: Unbounded

Answer:A: 1

Question: Let S T and U be nonempty sets and let f Sightarrow T and g Tightarrow U be functions such that the function gofSightarrow U is one to one injective Which of the following must be true

Question: Find limwrightarrow 2sqrt3w37w2

Answer:A: 2

Question: Find the limxrightarrow 0frac11efrac1x

Answer:A: 1

Question: Evaluate the limxrightarrow 1fracxx 1

Answer:A: 1

Question: Suppose fx gx exist g1xeq 0 and fxgx0 So limtightarrow xfracftgt

Answer:fracf1xg1x

Answer:f is one to one

Question: Let g be the function defined by gxe2x1 for all real x Then limxightarrow 0fracggx gex

Answer:4e2e2

Question: If f is a continuously differentiable real valued function defined on the open interval 1 4 such that f3 5 and f1x geq 1 for x what is the greatest possible value of f0

Question: Every nonempty set of real numbers of that has an upper bound also has a supremum in R This is the

Answer:The Completeness

Question: Find the derivatives of the function fxx2sinfrac1x when xeq 0 and fx0 when x0

Answer:0

Question: Using method of mathematical induction prove this 3 11 cdots 8n 5 for all nin N

Answer:n4 n

Answer:8

Question: limnightarrow 0 sinfrac1n

Answer:does not exist

Question: Evaluate limxightarrow 0fraccos3x 1x2

Answer: 92

Question: Evaluate limxrightarrow 0fraccos3x 1x2

Answer:D: 45

Question: limnrightarrow 0 sinfrac1n

Answer:D: does not exist

Question: Let g be the function defined by gxe2x1 for all real x Then limxrightarrow 0fracggx gex

Answer:C: 4e2e2

Question: What is the first derivative of the function fxxn

Answer:D: nxn 1

Question: Suppose fx gx exist g1xneq 0 and fxgx0 So limtrightarrow xfracftgt

Answer:B: fracf1xg1x

Question: Find the derivatives of the function fxx2sinfrac1x when xneq 0 and fx0 when x0

Answer:B: 0

Question: Let Sn be a convergent If limnightarrow inftySnS Then

Question: Let S T and U be nonempty sets and let f Srightarrow T and g Trightarrow U be functions such that the function gofSrightarrow U is one to one injective Which of the following must be true

Answer:A: f is one to one

Question: If S is a subset of R that contains at least two points and has the property if xyin S and xy then left xy ight subseteq S then S is an interval This is the Theorem

Answer:Characterization

Question: A sequence 1n is

Answer:Bounded

Answer:limnightarrow inftySn1S1

Question: Let limnightarrow inftyanmp The sum an convergent absolutely if p1 and it divergent if p 1 This theorem is called

Answer:ratio test theorem

Question: Solve limxightarrow 1fracx52x3

Answer:4

Question: If S is a subset of R that contains at least two points and has the property if xyin S and xy then left xy right subseteq S then S is an interval This is the Theorem

Question: Solve limxightarrow 2x24x

Answer:12

Question: The Bolzano Weierfrassthereom states that

Answer:A bounded sequences of real number has a convergent subsequences

Question: Do the series sumn1infty11n convergent or divergent

Answer:divergent

Question: A sequence left Sn ight is said to be bounded if

Answer:There exists positive real number s such that Sn s for all nin Z

Question: Let Sn be a convergent If limnrightarrow inftySnS Then

Answer:A limnrightarrow inftySn1S1

Question: Solve limxrightarrow 1fracx52x3

Answer:C 4

Question: Evaluate limxrightarrow 2fracx3 4x21

Answer:A 43

Answer:B Characterization

Question: A sequence left Sn right is said to be bounded if

Answer:C ratio test theorem

Answer:C There exists positive real number s such that Sn s for all nin Z

Question: Solve limxrightarrow 2x24x

Answer:B 12

Question: Let limnrightarrow inftyanmp The sum an convergent absolutely if p1 and it divergent if p 1 This theorem is called

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