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The value of _ x x^3 (1- ( 2 x ) )+3 x 4 x+1 ( x+3 x-1 )^ x 4 is :

Options

  1. A7 4 e
  2. B5 4 e
  3. C3 4 e
  4. D5 4 e ^4

Correct answer

B. 5 4 e

Step-by-step solution

We can separate the limit into a product of two limits: L₁ = _ x x^3 (1- ( 2 x ) )+3 x 4 x+1 and L₂ = _ x ( x+3 x-1 )^ x 4 For L₁ , we can rewrite the expression by multiplying and dividing the cosine term by ( 2 x )^2 : L₁ = _ x x 1- ( 2 x ) ( 2 x )^2 4 + 3x 4x+1 As x , 2 x 0 . We know that _ 0 1- ^2 = 1 2 . Substituting this into the limit: L₁ = _ x x ( 1 2 ) (4) + 3x 4x+1 = _ x 2x + 3x 4x+1 = _ x 5x 4x+1 = 5 4 For L₂ , we have a 1^ form: L₂ = _ x (1 + 4 x-1 )^ x 4 Using the formula _ x (1 + f(x))^ g(x) = e ^ _ x

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