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Continuity And Differentiability

Question
CBSEENMA12034600
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Find the relationship between a and b so that 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 a space straight x plus 1 comma space space space if space straight x less or equal than 3 end cell row cell straight b space straight x plus 3 comma space space space if space straight x greater than 3 end cell end table close
is continous at x=3

Solution
straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight a space straight x plus 1 comma space space space if space straight x less or equal than 3 end cell row cell straight b space straight x plus 3 comma space space space if space straight x greater than 3 end cell end table close space Lt with straight x rightwards arrow 3 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 3 to the power of minus below left parenthesis straight a space straight x plus 1 right parenthesis equals 3 space straight a plus 1 space Lt with straight x rightwards arrow 3 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 3 to the power of plus below left parenthesis straight b space straight x space plus space 3 right parenthesis equals space 3 space straight b plus 3 Also space straight f left parenthesis 3 right parenthesis equals 3 space straight a plus 1 Since space straight f space is space continous space at space straight x equals 3 therefore Lt with straight x rightwards arrow 3 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 3 to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 3 right parenthesis therefore space 3 straight a plus 1 equals 3 straight b plus 3 equals 3 space straight a plus 1 therefore space space space space space space 3 space straight a equals 3 straight b plus 2 rightwards double arrow space space space space space space space space straight a equals straight b plus 2 over 3 comma space which space is space required space relation.

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