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BITSAT2024MathematicsContinuity and DifferentiabilityActual

Let f be the function defined by f(x)= array cc x^2-1 x^2-2|x-1|-1 , & x 1 1 2 , & x=1 array .

Options

  1. AThe function is continuous for all values of x
  2. BThe function is continuous only for x 1
  3. CThe function is continuous at x=1
  4. DThe function is not continuous at x=1

Correct answer

D. The function is not continuous at x=1

Step-by-step solution

f(x)= array cc x^2-1 x^2-2 x+1 ; & x 1 1 2 ; & x=1 x^2-1 x^2+2 x-3 ; & x 1 array . x^2+3 x-x-3=x(x+3)-1(x+3) Lets Find LHL, _ x 1⁻ x^2-1 x^2-2 x+1 and RHL, _ x 1⁺ x^2-1 x^2+2 x-3 aligned & _ x 1⁻ (x-1)(x+1) (x-1)^2 _ x 1⁺ (x-1)(x+1) (x+3)(x-1) & _ x 1⁻ x+1 x-1 = _ x 1⁺ x+1 x+3 = 2 4 = 1 2 aligned LHL f(1) f(x) is not continuous at x=1

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