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Continuity and Differentiability

Short Answer Type

61. find the value of the k so that the given function 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 fraction numerator straight k space cos space straight x over denominator straight pi minus 2 straight x end fraction comma space if space straight x not equal to straight pi over 2 end cell row cell space space space space space space space 3 space space space space space space comma space if space straight x equals straight pi over 2 end cell end table close at space straight x equals straight pi over 2

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62. Find the value of a and b so that the function f given by

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 1 space space space comma space space space space space straight x less or equal than 3 end cell row cell ax plus straight b comma space space space space 3 less than straight x less than 5 end cell row cell 7 space space space comma space space space space straight x greater or equal than 5 end cell end table close

is continous at x=3 and x=5

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63. If the function

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell space space space 3 ax plus straight b space space if space straight x greater than 1 end cell row cell space space space space space space 11 space space space space space space space space if space straight x equals 1 end cell row cell space space 5 ax minus 2 straight b space space if space straight x less than 1. end cell end table close

is continous at x=1, find the values of a and b.

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64. For what value of k is the following function continuous at x = 2?

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell table row cell 2 straight x plus 1 comma space space space space straight x less than 2 end cell row cell space space space straight k space space space space comma space space space space space space straight x equals 2 end cell end table end cell row cell space 3 straight x minus 1 comma space space space space space straight x greater than 2 end cell end table close

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Long Answer Type

65. Given that

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell fraction numerator 1 minus cos space 4 straight x over denominator straight x squared end fraction space space space space space space space space space space comma space if space straight x less than 0 end cell row cell space space space space space space straight a space space space space space space space space space space space space space space space space space space space space space space space comma space if space straight x equals 0 end cell row cell fraction numerator square root of straight x over denominator square root of 16 plus square root of straight x end root minus 4 end fraction space space comma space if space straight x greater than 0 end cell end table close

If f(x) is continous at x=0, find the value of a.

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66. Determine the value of a, b_y c for which the functions

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell fraction numerator sin left parenthesis straight a plus 1 right parenthesis straight x plus sin space straight x over denominator straight x end fraction space space space comma space straight x less than 0 end cell row cell space space space space space space space space space space space space space space space space space space space space straight c space space space space space space space space space space space space space space space comma space straight x equals 0 end cell row cell fraction numerator square root of straight x plus bx squared end root minus square root of straight x over denominator bx to the power of begin display style 3 over 2 end style end exponent end fraction space space space space space space space comma space straight x greater than 0 end cell end table close

is continous at x=0.

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Short Answer Type

67. Is the function defined by f(x) = | x |, a continuous function ?

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68. Discuss the continuity of the function f given by f(x) = x³ + x ² –1.

Here f(x) = x³ + x² –1
Function f is defined for all real numbers.
Let c be any real number
$therefore space straight f left parenthesis straight c right parenthesis equals straight c cubed plus straight c squared minus 1 Also space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below left parenthesis straight x cubed plus straight x squared minus 1 right parenthesis equals straight c cubed plus straight c squared minus 1 therefore space Lt with straight x leftwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis$
∴ f is continuous at c
But c is any real number
∴ f is continuous at every real number
∴ f is continuous function.