Question

$If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell open vertical bar straight x close vertical bar sin 1 over straight x comma space if space straight x not equal to 0 end cell row cell space space space space space space space 0 comma space space space space space if space straight x equals 0 end cell end table close$
then discuss continuity of f(x) at x = 0.

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell open vertical bar straight x close vertical bar sin 1 over straight x comma space if space straight x not equal to 0 end cell row cell space space space space space space space 0 comma space space space space space if space straight x equals 0 end cell end table close space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of minus below open vertical bar straight x close vertical bar sin 1 over straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 0 to the power of minus right square bracket space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below open vertical bar 0 minus straight h close vertical bar space sin fraction numerator 1 over denominator 0 minus straight h end fraction space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below straight h space sin 1 over straight h equals 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets table row cell because Lt with straight x rightwards arrow 0 below straight h equals 0 and space sin 1 over straight h is space bounded space function space as space open vertical bar sin space 1 over straight h close vertical bar less or equal than 1 end cell row cell therefore Lt with straight h rightwards arrow 0 below straight h space sin 1 over straight h equals 0 space as space we space know space that space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis straight g left parenthesis straight x right parenthesis equals 0 end cell row cell if space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals 0 space and space straight g left parenthesis straight x right parenthesis space is space bounded space in space deleted space straight n space straight b space straight d space of space 0. end cell end table close square brackets

space Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below open vertical bar straight x close vertical bar space sin 1 over straight x space left square bracket Putting space straight x equals 0 plus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space rightwards arrow 0 to the power of plus right square bracket space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below open vertical bar 0 plus straight h close vertical bar sin fraction numerator 1 over denominator 0 plus straight h end fraction equals Lt with straight h rightwards arrow 0 below straight h space sin 1 over straight h equals 0 space space space space left square bracket As space explained space above right square bracket Also space straight f left parenthesis 0 right parenthesis equals 0 therefore Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis

∴ f(x) is continuous at x = 0.

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