Discuss the continuity or otherwise of the function f(x) defined
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    Continuity and Differentiability

    Multiple Choice QuestionsShort Answer Type

    21.

    Examine the  continuity of 
    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open vertical bar straight x minus straight a close vertical bar over denominator straight x minus straight a end fraction space space space space space space straight x not equal to straight a end cell row cell space space space space space space space space 1 space space space space space space space space space space space straight x equals straight a end cell end table close space at space space straight x equals straight a

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    22. Examine the continuity of f (x) at x = 0.
    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator 2 open vertical bar straight x close vertical bar end fraction comma space straight x not equal to 0 end cell row cell space space space space space 1 half comma space straight x equals 0 end cell end table close
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    23. Examine the continuity of f (x) at x = 0.
    If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x minus open vertical bar straight x close vertical bar comma space straight x not equal to 0 end cell row cell space space space space space space 2 space space space comma space straight x equals 0 end cell end table close
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    24.  Test the continuity of the function f(x) at the origin :
    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction comma space space space space straight x not equal to 0 end cell row cell space space space 1 space space space space comma space space space space straight x equals 0 end cell end table close
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    Multiple Choice QuestionsLong Answer Type

    25. Test the continuity of the function f(x) at the originstraight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction comma space space space space straight x not equal to 0 end cell row cell space space space 0 space space space space comma space space space space straight x equals 0 end cell end table close
    then show that f(x) is discontinuous at x = 0.
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    26. Prove that the function
    straight f left parenthesis straight x right parenthesis open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator open vertical bar straight x close vertical bar plus 2 straight x squared end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space straight k space space space space space space space comma space straight x equals 0 end cell end table close
    remains discontinuous at x = 0, regardless of the choice of k.
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    27. Show that the function f given by f(x) = | x | + | x –1 |,x ∈ R is continuous both at x =.0 and x = 1.
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    28.

    Discuss continuity of the function f given by

    f(x) = | x – 1| + | x – 2 ] at x = 1 and x = 2.

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    Multiple Choice QuestionsShort Answer Type

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    29. Discuss the continuity or otherwise of the function f(x) defined by
    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin 3 straight x over denominator straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space 1 space space space space space space comma space straight x equals 0 end cell end table close space at space straight x equals 0

    We space have space space space space space space space space space space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space 3 straight x over denominator straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space 1 space space space space space space comma space straight x equals 0 end cell end table close space space Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 below fraction numerator sin space 3 straight x over denominator straight x end fraction equals Lt with straight x rightwards arrow 0 below open square brackets 3. fraction numerator sin space 3 straight x over denominator 3 straight x end fraction close square brackets equals 3 Lt with straight x rightwards arrow 0 below fraction numerator sin space 3 straight x over denominator 3 straight x end fraction equals 3 cross times 1 equals 3 But space space straight f left parenthesis 0 right parenthesis equals 1 therefore Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis not equal to straight f left parenthesis 0 right parenthesis
    ⇒ f is not continuous at x = 0
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    30. Discuss the continuity or otherwise of the function f(x) defined by
    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space 2 straight x over denominator sin space 3 straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space space space 2 space space space space space comma space straight x equals 0 end cell end table close at space straight x equals 0
    88 Views
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