Test the continuity of the following function at the points indicated against them: f(x)=x3-8x+2-3x-2 for x ≠ 2 = – 24 for x = 2, at x = 2 - Mathematics and Statistics

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Sum

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

`"f"(x) = (x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))`  for x ≠ 2
         = – 24                               for x = 2, at x = 2

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Solution

f(2) = – 24   ...(given)

`lim_(x→2) "f"(x) = lim_(x→2) (x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))`

= `lim_(x→2) (x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2)) xx (sqrt(x + 2) + sqrt(3x - 2))/(sqrt(x + 2) + sqrt(3x - 2))`

= `lim_(x→2) ((x^3 - 8) (sqrt(x + 2) + sqrt(3x - 2)))/((x + 2) - (3x - 2))`

= `lim_(x→2) ((x^3 - 2^3)(sqrt(x + 2) + sqrt(3x - 2)))/(-2x + 4)`

= `lim_(x→) ((x - 2) (x^2 + 2x + 4) (sqrt(x + 2) + sqrt(3x - 2)))/(-2(x - 2))`

= `lim_(x→2) ((x^2 + 2x + 4)(sqrt(x + 2) + sqrt(3x - 2)))/-2  ...[(because x→ 2","  x ≠ 2),(therefore x- 2 ≠ 0)]`

= `(-1)/2 lim_(x→2) (x^2 + 2x + 4) (sqrt(x + 2) + sqrt(3x - 2))`

= `(-1)/2 lim_(x→2) (x^2 + 2x + 4) lim_(x→2)(sqrt(x + 2) + sqrt(3x - 2))`

= `(-1)/2 xx [2^2 + 2(2) + 4] xx (sqrt(2 + 2) + sqrt(3(2) - 2))`

= `(-1)/2 xx 12 xx (2 + 2)`

= – 24
∴ `lim_(x→2) "f"(x) = "f"(2)`
∴ f(x) is continuous at x = 2

Concept: Properties of Continuous Functions
  Is there an error in this question or solution?
Chapter 8: Continuity - Exercise 8.1 [Page 112]

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