Examine the continuity of f (x) at x = 0. from Mathematics Cont

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Continuity and Differentiability

Short Answer Type

21.

Examine the continuity of
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open vertical bar straight x minus straight a close vertical bar over denominator straight x minus straight a end fraction space space space space space space straight x not equal to straight a end cell row cell space space space space space space space space 1 space space space space space space space space space space space straight x equals straight a end cell end table close space at space space straight x equals straight a$

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22. Examine the continuity of f (x) at x = 0.
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator 2 open vertical bar straight x close vertical bar end fraction comma space straight x not equal to 0 end cell row cell space space space space space 1 half comma space straight x equals 0 end cell end table close$

Here 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 over denominator 2 open vertical bar straight x close vertical bar end fraction comma space straight x not equal to 0 end cell row cell space space space space space 1 half comma 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 space Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator straight x over denominator 2 open vertical bar straight x close vertical bar end fraction equals space Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator straight x over denominator 2 left parenthesis negative straight x right parenthesis end fraction space space space left square bracket because space open vertical bar straight x close vertical bar equals negative straight x space for space straight x less than 0 right square bracket space space space space space space space space space space space space space space space space equals space minus 1 half space Lt with straight x rightwards arrow 0 to the power of minus below equals negative 1 half space Lt with straight x rightwards arrow 0 to the power of minus below space left parenthesis 1 right parenthesis equals negative 1 half cross times 1 equals space minus 1 half space Lt with 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 fraction numerator straight x over denominator 2 open vertical bar straight x close vertical bar end fraction equals space Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator straight x over denominator 2 space straight x end fraction space space space space space space space space space left square bracket because space open vertical bar straight x close vertical bar equals straight x space for space straight x greater than 0 right square bracket space space space space space space space space space space space space space space space space equals 1 half Lt with straight x rightwards arrow 0 to the power of plus below left parenthesis 1 right parenthesis equals 1 half cross times 1 equals 1 half therefore space space space space space space space Lt with straight x rightwards arrow 0 to the power of minus below 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 straight f left parenthesis straight x right parenthesis space space space space space space space space space rightwards double arrow space space space straight f left parenthesis straight x right parenthesis space is space discontinous space at space straight x equals 0.

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23. Examine the continuity of f (x) at x = 0.

If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x minus open vertical bar straight x close vertical bar comma space straight x not equal to 0 end cell row cell space space space space space space 2 space space space comma space straight x equals 0 end cell end table close

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24. Test the continuity of the function f(x) at the origin :

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 close vertical bar over denominator straight x end fraction comma space space space space straight x not equal to 0 end cell row cell space space space 1 space space space space comma space space space space straight x equals 0 end cell end table close

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Long Answer Type

25. Test the continuity of the function f(x) at the origin

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 close vertical bar over denominator straight x end fraction comma space space space space straight x not equal to 0 end cell row cell space space space 0 space space space space comma space space space space straight x equals 0 end cell end table close

then show that f(x) is discontinuous at x = 0.

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26. Prove that the function

straight f left parenthesis straight x right parenthesis 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 plus 2 straight x squared end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space straight k space space space space space space space comma space straight x equals 0 end cell end table close

remains discontinuous at x = 0, regardless of the choice of k.

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27. Show that the function f given by f(x) = | x | + | x –1 |,x ∈ R is continuous both at x =.0 and x = 1.

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28.

Discuss continuity of the function f given by

f(x) = | x – 1| + | x – 2 ] at x = 1 and x = 2.

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Short Answer Type

29. Discuss the continuity or otherwise of the function f(x) defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin 3 straight x over denominator straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space 1 space space space space space space comma space straight x equals 0 end cell end table close space at space straight x equals 0

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30. Discuss the continuity or otherwise of the function f(x) defined 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 2 straight x over denominator sin space 3 straight x end fraction comma space straight x not equal to 0 end cell row cell 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

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