Question

Determine if f is defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x squared space sin 1 over straight x comma space straight x not equal to 0 end cell row cell space space 0 space space space space space comma space straight x equals 0 end cell end table close$

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Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x squared space s i n 1 over straight x comma space straight x not equal to 0 end cell row cell space space 0 space space space space space comma space straight x equals 0 end cell end table close space stack Lt space space space with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 0 to the power of minus below open parentheses straight x squared sin 1 half close parentheses space space space space space space space space space space space space left square bracket Put space straight x equals 0 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 space Lt with straight x rightwards arrow 0 below open curly brackets left parenthesis 0 minus straight h right parenthesis squared space sin 1 over straight h close curly brackets equals space Lt with straight x rightwards arrow 0 below straight h squared sin 1 over straight h space space space space space space space space space space space space space space space space space equals 0 space space space space space space space space space space space space space space space space space space space space space open square brackets because space Lt with straight h rightwards arrow 0 below straight h squared equals 0 space and space sin 1 over straight h space is space bounded space in space deleted space nbd space of space 0 close square brackets space stack space Lt with space space straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 0 to the power of plus below space straight x squared sin 1 over straight x equals space Lt with straight h rightwards arrow 0 below open curly brackets open parentheses 0 plus straight h close parentheses squared space sin fraction numerator 1 over denominator 0 plus straight h end fraction close curly brackets equals space Lt with straight x rightwards arrow 0 below straight h squared space sin 1 over straight h equals 0 Also space straight f left parenthesis 0 right parenthesis space space Lt with straight x rightwards arrow 0 to the power of minus below space straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals space straight f left parenthesis 0 right parenthesis

∴ f(x) is continuous at x = 0

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