Question

Examine the continuity of the following function :
f(x) = x² - x + 9,          for x ≤ 3
      = 4x + 3,               for x > 3 
at x = 3.

Answer 1

Given :
f(x) = x² - x + 9,          for x ≤ 3
      = 4x + 3,               for x > 3 
at 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)`

Hense function is continuous at x = 3.