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Continuity and Differentiability

Short Answer Type

11. Examine the continuity of

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x squared minus 1 over denominator straight x plus 1 end fraction comma space straight x not equal to negative 1 end cell row cell space space minus 2 space space space space space comma space straight x equals negative 1 end cell end table close

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12. Show that the function f given by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x cubed plus 3 comma space if space straight x not equal to 0 end cell row cell 1 comma space space space space space space space space space if space straight x not equal to 0 end cell end table close

is not continuous at x = 0.

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13. find whether f is continuous at x = 1.

space If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x squared minus 1 over denominator straight x minus 1 end fraction comma space straight x not equal to 1 end cell row cell 2 space space space space space space space space space space space comma straight x equals 1 end cell end table close

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14. find whether f is continuous at x = 3

If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x squared minus 9 over denominator straight x minus 3 end fraction comma space straight x not equal to 3 end cell row cell space space space space space space 6 space space space space space comma straight x equals 3 end cell end table close

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15.

Show that the function is continuous at x = 3.
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x squared minus 7 straight x plus 12 over denominator straight x minus 3 end fraction comma space straight x space not equal to 3 end cell row cell space space space space space space space space space space minus 1 space space space space space space space space comma space straight x equals 3 end cell end table close$

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16. Discuss continuity of function at x = 2, where

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 2 straight x squared 3 straight x minus 2 over denominator straight x minus 2 end fraction comma space straight x not equal to 2 end cell row cell space space space space space space space space space 3 space space space space space space space space space comma space straight x equals 2 end cell end table close

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17. A function f is defined as

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x squared minus straight x minus 6 over denominator straight x squared minus 2 straight x minus 3 end fraction comma space space straight x not equal to 3 end cell row cell space space space space space space space space space 5 over 3 space space space space space space space space comma space straight x equals 3 end cell end table close

Prove that f is discontinuous at x = 3. Can the definition of f at x = 3 be modified so as to make it continuous there?

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18. Test the continuity of the function at x = 3, where

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open vertical bar straight x minus 3 close vertical bar over denominator straight x left parenthesis straight x minus 3 right parenthesis end fraction space space comma space space straight x not equal to 3 end cell row cell space space space space space space space 0 space space space space space space space space space comma space space straight x equals 3 end cell end table close

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19.

Is $straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open square brackets straight x close square brackets minus 1 over denominator straight x minus 1 end fraction comma space straight x not equal to 1 end cell row cell space space space minus 1 space space space space space space comma space straight x equals 1 end cell end table close$ continous at x=1 ?

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20. Examine the continuity of
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction comma space if space straight x not equal to 0 end cell row cell space space space space 0 space space space comma space if space straight x equals 0 end cell end table close space space space space space space space space space at space straight x equals 0$

Here space straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction comma space if space straight x not equal to 0 end cell row cell space space space space 0 space space space comma 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 space straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction 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 plus 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 space space space space space space space space space equals space Lt with straight h rightwards arrow 0 to the power of minus below fraction numerator 0 plus straight h over denominator open vertical bar 0 plus straight h close vertical bar end fraction equals space Lt with straight h rightwards arrow 0 below fraction numerator straight h over denominator open vertical bar straight h close vertical bar end fraction equals space Lt with straight h rightwards arrow 0 below space straight h over straight h space equals space Lt with straight h rightwards arrow 0 below space space space 1 equals 1 therefore space Lt with straight x rightwards arrow 0 to the power of minus below space straight f left parenthesis straight x right parenthesis space not equal to space Lt with straight x rightwards arrow 0 to the power of plus below therefore straight f space is space continous space at space straight x equals 0. space space space space space space space

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