Question

If f(x) is continuous at .x = 0, find a, where
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space ax over denominator straight x space sinx end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space space space space 1 half space space space space comma space straight x equals 0 end cell end table close$

Solution

We space have 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 1 minus cos space ax over denominator straight x space sinx end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space space space space 1 half space space space space comma space straight x equals 0 end cell end table close 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 1 minus cos space ax over denominator straight x space sin space straight x end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator 1 minus cos space ax over denominator straight x squared begin display style fraction numerator sin space straight x over denominator straight x end fraction end style end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator 1 minus cos space ax over denominator straight x squared end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator 2 sin squared begin display style ax over 2 end style over denominator straight x squared end fraction space space space space space space space equals straight a squared over 2 Lt with straight x rightwards arrow 0 below open parentheses fraction numerator sin begin display style ax over 2 end style over denominator begin display style ax over 2 end style end fraction close parentheses squared equals straight a squared over 2 left parenthesis 1 right parenthesis squared equals straight a squared over 2 Since space straight f left parenthesis straight x right parenthesis space is space continous space at space straight x equals 0 therefore space Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis space space rightwards double arrow space straight a squared over 2 equals 1 half space space rightwards double arrow straight a squared equals 1 therefore space straight a equals plus-or-minus 1.

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