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MHT CET202613 April 2026Evening ShiftMathematicsContinuity and DifferentiabilityActual

If f(x) is continuous at x = 0 , where f(x) = 8^x - 2^x k^x - 1 , for x 0 and f(0) = 2 , then the value of k is ...

Options

  1. A0
  2. B4
  3. C-2
  4. D2

Correct answer

D. 2

Step-by-step solution

Since f(x) is continuous at x = 0 , we must have _ x 0 f(x) = f(0) . _ x 0 8^x - 2^x k^x - 1 = 2 Dividing the numerator and the denominator by x : _ x 0 8^x - 1 x - 2^x - 1 x k^x - 1 x = 2 Using the standard limit _ x 0 a^x - 1 x = a : 8 - 2 k = 2 ( 8 2 ) k = 2 4 k = 2 4 = 2 k 4 = (k^2) k^2 = 4 Since k > 0 for the exponential function k^x to be defined for all real x , we get k = 2 .

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