Question: Let fRR be defined by fxx2 then the direct image of the set
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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