Question

Examine the continuity of f(x)=`x^2-x+9  "for"  x<=3`

=`4x+3  "for"  x>3,  "at"  x=3` 

Answer 1

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

`lim_(x -> 3^-)` f(x) = `lim_(x -> 3)` (x² - x + 9)

= 9 - 3 + 9

= 15

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

= (4 x 3) + 3

= 15 .........(i)

f(3) = 3² - 3 + 9

= 15.......(ii)

From (i) and (ii)

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

`therefore` Function f(x) is continuous at x = 3.