Question

Check whether the function f(x) is defined as:

`f(x) = {((|x - 3|)/(2(x - 3))","  x < 3), ((x - 6)/6","  x ≥ 3):}` is continuous at x = 3 or not?

Answer 1

For x < 3, ∣x − 3∣ = 3 − x

`lim_(x→3^-) f(x) = lim_(x→3^-) (3 - x)/(2(x - 3)) = -1/2`

For x ≥ 3,

`f(3) = (3 - 6)/6`

`f(3) = -1/2`

Also,

`lim_(x→3^+) f(x) = lim_(x→3^+) (x - 6)/6 = -1/2`

Since LHL = RHL = f(3), f(x) is continuous at x = 3.