Question

Which of the following functions are continuous at the indicated points? $straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell space space 5 straight x minus 4 space space space comma space if space straight x less or equal than 1 end cell row cell 4 straight x squared minus 3 straight x space space comma space if space straight x greater than 1 end cell end table close$

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell space space 5 straight x minus 4 space space space comma space if space straight x less or equal than 1 end cell row cell 4 straight x squared minus 3 straight x space space comma space if space straight x greater than 1 end cell end table close space Lt with straight x rightwards arrow 1 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 left parenthesis 5 straight x minus 4 right parenthesis 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 1 minus 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 equals Lt with straight h rightwards arrow 0 below left curly bracket 5 left parenthesis 1 minus straight h right parenthesis minus 4 right curly bracket equals Lt with straight h rightwards arrow 0 below left parenthesis 1 minus 5 straight h right parenthesis equals 1 minus 0 equals 1 space Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of plus below left parenthesis 4 straight x squared minus 3 straight x right parenthesis 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 1 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 equals Lt with straight h rightwards arrow 0 below left curly bracket 4 left parenthesis 1 plus straight h squared right parenthesis minus 3 left parenthesis 1 plus straight h right parenthesis right curly bracket equals Lt with straight h rightwards arrow 0 below left parenthesis 4 straight h squared plus 8 straight h plus 4 minus 3 minus 3 straight h right parenthesis 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 left parenthesis 4 straight h squared plus 5 straight h plus 1 right parenthesis equals 0 plus 0 plus 1 equals 1 Also space space straight f left parenthesis 1 right parenthesis equals Value space of left parenthesis 5 straight x minus 4 right parenthesis space at space straight x equals 1 equals 5 minus 4 equals 1 therefore space Lt with straight x rightwards arrow 1 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of minus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 1 right parenthesis

∴ function f(x) is continuous at x = 1.

For Daily Free Study Material Join wiredfaculty

Continuity And Differentiability

Some More Questions From Continuity and Differentiability Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Location