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COMEDK2023Morning ShiftMathematicsLimitsActual

The value of _ x ( x^2-2 x+1 x^2-4 x+2 )^ 2 x is

Options

  1. Ae^2
  2. Be¹⁶
  3. Ce
  4. De^4

Correct answer

D. e^4

Step-by-step solution

The limit is of the form 1^ as x . Let L = _ x ( x^2-2x+1 x^2-4x+2 )^ 2x . Using the formula _ x a [f(x)]^ g(x) = e^ _ x a g(x)[f(x)-1] , we have: L = e^ _ x 2x ( x^2-2x+1 x^2-4x+2 - 1 ) L = e^ _ x 2x ( x^2-2x+1 - (x^2-4x+2) x^2-4x+2 ) L = e^ _ x 2x ( 2x-1 x^2-4x+2 ) L = e^ _ x 4x^2-2x x^2-4x+2 Dividing numerator and denominator by x^2 : L = e^ _ x 4 - 2/x 1 - 4/x + 2/x^2 = e^ 4/1 = e^4 . Answer: e^4

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