# Test the continuity of the following function at the points indicated against them: f(x)=x-1-(x-1)13x-2 for x ≠ 2 = 15 for x = 2, at x = 2 - Mathematics and Statistics

Sum

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

Concept: Properties of Continuous Functions
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