Question

Examine the continuity of the following function :

`{:(,,f(x)= x^2 -x+9,"for",x≤3),(,,=4x+3,"for",x>3):}}"at "x=3`

Answer 1

Given

`f(x)=x^2-x+9 , for x<=3`

` =4x+3        for x>3`

`f(3)=(3)^2-3+9=9-3+9`

`f(3)=15`

`Now lim_(x->3^-)f(x)=lim_(x->3)(x^2-x+9)`

                         ` =(3)^2-(3)+9`

                           =15

`lim_(x->3^-)f(x)=lim_(x->3)(4x+3)`

                        =4(3)+3

                        =15

Thus from the above 

`lim_(x->3^-)f(x)=lim_(x->3)f(x)=15=f(3)`

Hence function is continuous at x=3