Find the value of k so that the function f is continuous at the

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Continuity and Differentiability

Short Answer Type

101. Find the value of k so that the function f is continuous at the indicated point

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell space straight k space straight x squared comma space straight x greater or equal than 1 end cell row cell space 4 space space space space comma space straight x less than 1 end cell end table close at space straight x equals 1

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102. Find the value of k so that the function f is continuous at the indicated point

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight k space straight x squared comma space straight x less or equal than 2 end cell row cell 3 space space space space comma space straight x greater than 2 end cell end table close at space straight x equals 2

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103. Find the value of k so that the function f is continuous at the indicated point

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight k space straight x plus 1 space space comma space straight x less or equal than 5 end cell row cell 3 x minus 5 space space space comma space straight x greater than 5 end cell end table close at space straight x equals 5

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104. Find the value of k so that the function f is continuous at the indicated point
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight k left parenthesis straight x squared minus 2 straight x space right parenthesis comma space if space straight x less than 0 end cell row cell space space cos space straight x space space space space space comma space if space straight x greater or equal than 0 end cell end table close at space straight x equals 0$

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight k left parenthesis straight x squared minus 2 straight x space right parenthesis comma space if space straight x less than 0 end cell row cell space space cos space straight x space space space space space comma space if space straight x greater or equal than 0 end cell end table close space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of minus below left curly bracket straight k left parenthesis straight x squared minus 2 straight x right parenthesis right curly bracket space space space space space space space space space space space space space space space space space space left square bracket Put space straight x equals 0 minus straight h comma space straight h greater than 0 right square bracket space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below straight k left curly bracket left parenthesis 0 minus straight h right parenthesis squared minus 2 left parenthesis 0 minus straight h right parenthesis right curly bracket equals straight k Lt with straight h rightwards arrow 0 below left parenthesis straight h squared plus 2 straight h right parenthesis space space space space space space space space space space space space space space space space equals straight k left parenthesis 0 plus 0 right parenthesis equals 0 Also space straight f left parenthesis 0 right parenthesis equals cos space 0 equals 1 therefore space Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis not equal to straight f left parenthesis 0 right parenthesis therefore space straight f left parenthesis straight x right parenthesis space is space not space continous space for space any space value space of space straight k.

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105. Discuss the continuity of the function f defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x plus 2 comma space if space straight x less or equal than 1 end cell row cell straight x minus 2 comma space if space straight x greater than 1 end cell end table close

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106. Find all the points of discontinuity of the function f defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell straight x plus 2 comma space if space straight x less than 1 end cell row cell space space space space 0 comma space space space if space straight x equals 1 end cell row cell straight x minus 2 comma space if space straight x greater than 1 end cell end table close

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107. Discuss the continuity of the function defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x plus 2 comma space space space space space if space straight x less than 0 end cell row cell negative straight x plus 2 comma space if space straight x greater than 0 end cell end table close

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108. Discuss the continuity of the function f given by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell space space straight x comma space if space straight x greater or equal than 0 end cell row cell straight x squared comma space if space straight x less than 0 end cell end table close

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109. Find all points of discontinuity of f, where f is defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 straight x plus 3 comma space if space straight x less or equal than 2 end cell row cell 2 straight x minus 3 comma space if space straight x greater than 2 end cell end table close

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Long Answer Type

110. Find all points of discontinuity of f, where f is defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell open vertical bar straight x close vertical bar plus 3 comma space if space straight x less or equal than negative 3 end cell row cell negative 2 straight x comma if space minus 3 less than straight x greater than 3 end cell row cell 6 straight x plus 2 space space comma space if space straight x greater or equal than 3 end cell end table close

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