Question

Determine if f define by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space straight x over denominator straight x end fraction plus space cos space straight x comma space straight x not equal to 0 end cell row cell space space space space space space space space space space 2 space space space space space comma space straight x equals 0 end cell end table close at space straight x equals 0$

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space straight x over denominator straight x end fraction plus space cos space straight x comma space straight x not equal to 0 end cell row cell space space space space space space space space space space 2 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 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 parentheses fraction numerator sin space straight x over denominator straight x end fraction plus space cos space straight x close parentheses equals Lt with straight x rightwards arrow 0 below fraction numerator sin space straight x over denominator straight x end fraction plus Lt with straight x rightwards arrow 0 below c os space straight x equals 1 plus 1 equals 2 Also space straight f left parenthesis 0 right parenthesis equals 2 therefore Lt with straight x rightwards arrow 0 below space straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis equals 2 comma space which space is space finite therefore space straight f left parenthesis straight x right parenthesis space is space contionus space at space straight x equals 0.

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