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Question
f(x) = `(sqrt(x + 3) - 2)/(x^3 - 1)` for x ≠ 1
= 2 for x = 1, at x = 1.
Solution
f(1) = 2 …[given]
`lim_(x→1) "f"(x) = lim_(x→1) (sqrt(x + 3) - 2)/(x^3 - 1)`
= `lim_(x→1) ((sqrt(x + 3) - 2)/(x^3 - 1) xx (sqrt(x + 3) + 2)/(sqrt(x + 3) + 2))`
= `lim_(x→1) ((x + 3 - 4)/((x^3 - 1)(sqrt(x + 3) + 2)))`
= `lim_(x→1) (x - 1)/((x - 1)(x^2 + x + 1)(sqrt(x + 3) + 2))`
= `lim_(x→1) 1/((x^2 + x + 1)(sqrt(x + 3) + 2)) ...[("As" x→1"," x ≠ 1),(therefore x - 1 ≠ 0)]`
= `1/(lim_(x→1)(x^2 + x + 1) xx lim_{x→1} (sqrt(x + 3) + 2))`
= `1/((1^2 + 1 + 1) xx (sqrt(1 + 3) + 2))`
= `1/(3(2 + 2))`
= `1/12`
∴ `lim_(x→1) "f"(x) ≠ "f"(1)`
∴ f is discontinuous at x = 1
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