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# MTH 241 TMA Questions & Answers

`  Question: Let fRR be defined by fxx2 then the direct image of the setAnswer:A: fEy0y4Question: Let f and g be the functions whose values at xR are given by fx2x and gx3x2 1The composite function gcirc f isAnswer:A:  gcirc fx12x2 1Question: Determine the set A of all real numbers x such that 2x36Answer:A: x32Question: If In0 frac1n for nin mattabN then In subseteq In1 for each nnin mattabN so that this sequence of intervals is Answer:A: NestedQuestion: 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 exceptAnswer:A: PeriodicQuestion: The unit interval 0 1  xR0x1 is Answer:A: Not countableQuestion: If  C is an infinite set and B is a finite set then CB is an Answer:A: infinite setAnswer:A: If tU then t is not an upper bound of SAnswer:A: frac1n2 n1leq frac1n2 Question: If xn  is an unbounded increasing sequence then limxn isAnswer:A:  infty Question: Consider the sequence by 1nn0infty sumn0ifty  1n1 11Answer:A: DivergentQuestion: The series sumn1infty frac1n2 n1 is convergent if the inequality Question: The compliment of B relative to A is the setQuestion:  If A123 andB45 then the set AB is the set whose elements are the ordered pairsAnswer:A: ABxxA and xBQuestion: If A B and C are sets then ABC is equal toAnswer:A: ABACQuestion: 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: ABabaAbBAnswer:A: 14 15 24 25 3435Question: A nonempty subset p of  is called the set of positive real numbers the following are the order properties of  exceptQuestion: A function meant a definite formula such as fxx23x50 which associates to each real number x another number fx then find f 4Answer:A: 9Question: A function which is both one one and onto is known asAnswer:A: BijectiveQuestion:  In the distributive law A BC is equal to Answer:A: AB  ACAnswer:A: If abPab1Question: Let a b c  be any elements of  the following are equivalent exceptQuestion: Consider the series sumn1inftyfrac1nn1frac112frac123frac134  this series converges toAnswer:A: If ab and c0 then cacbQuestion: The limit of the sequence frac1n21 is Answer:A: 1Question: The limit of the sequence frac2nn2 is Answer:A: 2Question: The Bolzano Weierstrass theorem say that a bounded sequence of real numbers has a Answer:A: Convergent subsequenceQuestion: A monotone sequence of real number is properly divergent if and only if it is Answer:A: UnboundedAnswer:A: 1Question: 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 trueQuestion: Find limwrightarrow  2sqrt3w37w2Answer:A: 2Question: Find the limxrightarrow 0frac11efrac1xAnswer:A: 1Question: Evaluate the limxrightarrow 1fracxx 1Answer:A: 1Question: Suppose fx gx exist g1xeq 0 and fxgx0 So limtightarrow xfracftgtAnswer:fracf1xg1xAnswer:f is one  to  oneQuestion: Let g be the function defined by gxe2x1 for all real x Then limxightarrow 0fracggx gexAnswer:4e2e2Question: 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 f0Question: Every nonempty set of real numbers of that has an upper bound also has a supremum in R This is the Answer:The CompletenessQuestion: Find the derivatives of the function fxx2sinfrac1x when xeq 0 and fx0 when x0Answer:0Question: Using method of mathematical induction prove this  3  11  cdots  8n 5   for all nin NAnswer:n4 nAnswer:8Question: limnightarrow 0 sinfrac1n Answer:does not existQuestion: Evaluate limxightarrow 0fraccos3x 1x2Answer: 92Question: Evaluate limxrightarrow 0fraccos3x 1x2Answer:D:  45Question: limnrightarrow 0 sinfrac1n Answer:D: does not existQuestion: Let g be the function defined by gxe2x1 for all real x Then limxrightarrow 0fracggx gexAnswer:C: 4e2e2Question: What is the first derivative of the function fxxnAnswer:D: nxn 1Question: Suppose fx gx exist g1xneq 0 and fxgx0 So limtrightarrow xfracftgtAnswer:B: fracf1xg1xQuestion: Find the derivatives of the function fxx2sinfrac1x when xneq 0 and fx0 when x0Answer:B: 0Question: Let Sn be a convergent If limnightarrow inftySnS ThenQuestion: 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 trueAnswer:A: f is one  to  oneQuestion: 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  TheoremAnswer:CharacterizationQuestion: A sequence 1n isAnswer:BoundedAnswer:limnightarrow inftySn1S1Question: Let limnightarrow inftyanmp  The sum an convergent absolutely if p1 and it divergent if p  1 This theorem is calledAnswer:ratio test theoremQuestion: Solve limxightarrow 1fracx52x3Answer:4Question: 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  TheoremQuestion: Solve  limxightarrow 2x24xAnswer:12Question: The Bolzano Weierfrassthereom states thatAnswer:A bounded sequences of real number has a convergent subsequencesQuestion: Do the series  sumn1infty11n convergent or divergentAnswer:divergentQuestion: A sequence left  Sn ight  is said to be bounded ifAnswer:There exists positive real number s such that  Sn  s for all nin ZQuestion: Let Sn be a convergent If limnrightarrow inftySnS ThenAnswer:A limnrightarrow inftySn1S1Question: Solve limxrightarrow 1fracx52x3Answer:C 4Question: Evaluate limxrightarrow 2fracx3 4x21Answer:A 43Answer:B CharacterizationQuestion: A sequence left  Sn right  is said to be bounded ifAnswer:C ratio test theoremAnswer:C There exists positive real number s such that  Sn  s for all nin ZQuestion: Solve  limxrightarrow 2x24xAnswer:B 12Question: Let limnrightarrow inftyanmp  The sum an convergent absolutely if p1 and it divergent if p  1 This theorem is called`

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