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Question
Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:
f(x) `{:(= (x^2 + x + 1)/(x + 1)"," , "for" x ∈ [0, 3)),(=(3x +4)/(x^2 - 5)"," , "for" x ∈ [3, 6]):}`
Solution
For x ∈ [0, 3), f(x) = `(x^2 + x + 1)/(x + 1)`, being a rational function is continuous except when its denominator x + 1 = 0 i.e., at x = – 1, which does not belong to [0, 3)
∴ f is continuous on [0, 3).
For x ∈ [3, 6], f(x) = `(3x + 4)/(x^2 - 5)`, being a rational function is continuous except when its denominator x2 – 5 = 0 i.e., at x = `± sqrt(5)` But `± sqrt(5) ∉ [3, 6]`
∴ f is continuous on [0, 6] except possibly at x = 3
Continuity at x = 3
f(x) = `(x^2 + x + 1)/(x + 1)`, for x ∈ [0, 3)
∴ `lim_(x -> 3^-) "f"(x) = lim_(x -> 3) (x^2 + x + 1)/(x + 1)`
= `(lim_(x -> 3)(x^2 + x + 1))/(lim_(x -> 3) (x + 1))`
= `(9 + 3 + 1)/(3 + 1)`
= `13/4`
Also, f(x) = `(3x + 4)/(x^2 - 5)`, for x ∈ [3, 6]
∴ `lim_(x -> 3^+) "f"(x) = "f"(3) = (9 + 4)/(9 - 5) = 13/4`
∴ `"f"(3) = lim_(x -> 3^+) "f"(x) = lim_(x -> 3^-) "f"(x)`
∴ f is continuous at x = 3.
Hence, f is continuous on its domain [0, 6].
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