Question

Test the continuity of the following function at the points indicated against them:

f(x) = 4x + 1,              for x ≤ 3

      = `(59 - 9x)/3`,        for x > 3 at x = `8/3`.

Answer 1

`lim_(x→(8/3)^-) "f"(x) = lim_(x→(8/3)^-) (4x + 1)`

= `4(8/3) + 1`

= `32/3 + 1`

= `35/3`

`lim_(x→(8/3)^+) "f"(x) = lim_(x→(8/3)^+) (59 - 9x)/3`

= `(59 - 9(8/3))/3`

= `(59 - 24)/3`

= `35/3`

f(x) = 4x + 1,      x ≤ `(8/3)`

∴ `"f"(8/3) = 4(8/3) + 1`

= `32/3 + 1`

= `35/3`

`lim_(x→(8/3)^-) "f"(x) = lim_(x→(8/3)^+) "f"(x) = "f"(8/3)`

∴ f(x) is continuous at x = `8/3`