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Question
Test the continuity of the following function at the points indicated against them:
`f(x) = (sqrt(x - 1) - (x - 1)^(1/3))/(x - 2)` for x ≠ 2
= `1/5` for x = 2, at x = 2
Solution
f(2) = `1/5` ...(given)
`lim_(x→2) "f"(x) = lim_(x→2) (sqrt(x - 1) - (x - 1)^(1/3))/(x - 2)`
Put x - 1 = y
∴ x = 1 + y
∴ As x → 2, y → 1
∴ `lim_(x→2) "f"(x) = lim_(y→1) (sqrt(y) - y^(1/3))/(1 + y - 2)`
= `lim_(y→1) (y^(1/2) - 1 - y^(1/3) + 1)/(y - 1)`
= `lim_(y→1) ((y^(1/2) - 1)-(y^(1/3) - 1))/(y - 1)`
= `lim_{y→1} ((y^(1/2) - 1)/(y - 1)- (y^(1/3) - 1)/(y - 1))`
= `lim_(y→1) (y^(1/2) - 1^(1/2))/(y - 1) - lim_{y→1} (y^(1/3) - 1^(1/3))/(y - 1)`
= `1/2(1)^((-1)/2) - 1/3(1)^((-2)/3)` ...[∵ `lim_(x→"a") (x^n - "a"^n)/(x - "a") = "n.a"^"n-1"`]
= `1/2 - 1/3`
= `1/6`
∴ `lim_(x→2) "f"(x) ≠ "f"(2)`
∴ f(x) is discontinuous at x = 2
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