Question

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

Solution

Put space sin to the power of negative 1 space end exponent straight x equals straight theta space or space straight x equals sin space straight theta space so space that space straight theta rightwards arrow 0 space space therefore Lt with straight x rightwards arrow 0 below open square brackets fraction numerator square root of 1 plus straight x end root minus square root of 1 minus straight x end root over denominator sin to the power of negative 1 end exponent end fraction close square brackets equals Lt with straight theta rightwards arrow 0 below open square brackets fraction numerator square root of 1 plus sin space straight theta end root minus square root of 1 minus sin space straight theta end root over denominator straight theta end fraction close square brackets equals Lt with straight theta rightwards arrow 0 below open square brackets fraction numerator square root of cos squared straight theta over 2 plus sin squared space straight theta over 2 plus 2 sin straight theta over 2 cos straight theta over 2 end root minus square root of cos squared begin display style straight theta over 2 end style plus sin squared begin display style straight theta over 2 end style minus 2 sin begin display style straight theta over 2 end style cos begin display style straight theta over 2 end style end root over denominator straight theta end fraction close square brackets equals Lt with straight theta rightwards arrow 0 below open square brackets square root of fraction numerator open parentheses cos straight theta over 2 plus sin straight theta over 2 close parentheses squared minus square root of open parentheses cos straight theta over 2 minus sin straight theta over 2 close parentheses squared end root over denominator straight theta end fraction end root close square brackets equals Lt with straight theta rightwards arrow 0 below open square brackets fraction numerator open parentheses cos begin display style straight theta over 2 end style plus sin begin display style straight theta over 2 end style close parentheses minus open parentheses cos begin display style straight theta over 2 end style minus sin begin display style straight theta over 2 end style close parentheses over denominator straight theta end fraction close square brackets space space space space space space space space space space space equals Lt with straight theta rightwards arrow 2 below fraction numerator 2 sin begin display style straight theta over 2 end style over denominator straight theta end fraction equals Lt with straight theta over 2 rightwards arrow 0 below open curly brackets fraction numerator sin begin display style straight theta over 2 end style over denominator begin display style straight theta over 2 end style end fraction close curly brackets equals 1 Also space straight f left parenthesis 0 right parenthesis equals 1 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

∴ f is continuous at x = 0.

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