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Mathematics : 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

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