Question

Which of the following functions are continuous at the indicated points ? $straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 straight x minus 1 space space space comma space space space if space space space straight x less than 2 end cell row cell space space space space space space 3 over 2 space space space space comma space space space if space space space straight x greater or equal than 2 end cell end table close at space straight x equals 2$

Solution

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 straight x minus 1 space space space comma space space space if space space space straight x less than 2 end cell row cell space space space space space 3 over 2 space space space comma space space space if space space space straight x greater or equal than 2 end cell end table close space Lt with straight x rightwards arrow 2 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 2 to the power of asterisk times minus end exponent below left parenthesis 2 straight x minus 1 right parenthesis space space space space space space space space space space space space space space space space space 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 2 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 space equals Lt with straight h rightwards arrow 0 below left curly bracket 2 left parenthesis 2 minus straight h right parenthesis minus 1 right curly bracket equals Lt with straight h rightwards arrow 0 below left parenthesis 4 minus 2 straight h minus 1 right parenthesis space 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 left parenthesis 3 minus 2 straight h right parenthesis space space space space space space space space space space space space space space space space space equals 3 minus 0 equals 3 Lt with straight x rightwards arrow 2 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 2 to the power of plus below fraction numerator 3 straight x over denominator 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space 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 2 plus 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 open curly brackets 3 over 2 left parenthesis 2 plus straight h right parenthesis close curly brackets equals 3 over 2 left parenthesis 2 plus 0 right parenthesis equals 3 Also space straight f left parenthesis 2 right parenthesis equals 3 over 2 left parenthesis 2 right parenthesis equals 3 therefore space Lt with straight x rightwards arrow 2 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 2 to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 2 right parenthesis

∴ f(x) is continuous at x = 2

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