Question

Discuss continuity of f(x) at x = 0, if
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator log left parenthesis 1 plus ax right parenthesis minus log left parenthesis 1 minus bx right parenthesis over denominator straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space space space space space space space space space space space straight a plus straight b space space space space space space space space space space space space space space space space space space space comma space straight x equals 0 end cell end table close$

Solution

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 open square brackets 1 over straight x minus fraction numerator log left parenthesis 1 plus straight x right parenthesis over denominator straight x squared end fraction close square brackets equals Lt with straight x rightwards arrow 0 below fraction numerator straight x minus log left parenthesis 1 plus straight x right parenthesis over denominator straight x squared end fraction space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below fraction numerator straight x minus open square brackets straight x minus begin display style straight x over 2 squared end style plus begin display style straight x cubed over 3 end style minus begin display style straight x to the power of 4 over 4 end style plus..... close square brackets over denominator straight x squared end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator begin display style straight x over 2 squared minus plus plus.... straight x cubed over 3 straight x to the power of 4 over 4 end style over denominator blank end fraction space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below fraction numerator straight x squared open square brackets begin display style 1 half plus terms space containing space straight x space and space its space higher space powers end style close square brackets over denominator straight x squared end fraction space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below open square brackets 1 half plus terms space containing space straight x space and space its space higher space powers close square brackets space space space space space space space space space space space space space space equals 1 half plus 0 equals 1 half Also space space straight f left parenthesis 0 right parenthesis equals 1 half therefore space straight f space is space continous space at space straight x equals 0 straight a

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