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

Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Advertisement

Continuity and Differentiability

Short Answer Type

21.

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

129 Views

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

79 Views

Advertisement

23. Examine the continuity of f (x) at x = 0.
$If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x minus open vertical bar straight x close vertical bar comma space straight x not equal to 0 end cell row cell space space space space space space 2 space space space comma space straight x equals 0 end cell end table close$

Here 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 space Lt with straight x rightwards arrow 0 to the power of minus below space straight f vertical line left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of minus below open curly brackets straight x minus open vertical bar straight x close vertical bar close curly brackets space equals Lt with straight x rightwards arrow 0 to the power of minus below open curly brackets left parenthesis 0 minus straight h right parenthesis minus open vertical bar 0 minus straight h close vertical bar close curly brackets 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 space space left square bracket space Put space straight x equals space 0 minus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 to the power of minus right square bracket space space space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below space open curly brackets negative straight h minus open vertical bar straight h close vertical bar close curly brackets space equals Lt with straight h rightwards arrow 0 below space open curly brackets negative straight h minus straight h close curly brackets equals negative 0 minus 0 equals 0 But space straight f left parenthesis 0 right parenthesis equals 2 therefore space Lt with straight x rightwards arrow 0 to the power of minus below space straight f left parenthesis straight x right parenthesis not equal to straight f left parenthesis 0 right parenthesis space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space straight f left parenthesis straight x right parenthesis space is space discontinous space at space straight x equals 0.

80 Views

Advertisement

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

89 Views

Advertisement

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.

77 Views

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.

174 Views

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

77 Views

28.

Discuss continuity of the function f given by

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

89 Views

Advertisement

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

87 Views

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

88 Views

2
3
4

Advertisement