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JEE Main202226 Jul 2022Evening ShiftMathematicsLimitsActual

Let β = lim x → 0 α x - e 3 x - 1 α x e 3 x - 1 for some α ∈ ℝ . Then the value of α + β is:

Options

  1. A14 5
  2. B3 2
  3. C5 2
  4. D7 2

Correct answer

C. 5 2

Step-by-step solution

Given, lim x → 0 α x - e 3 x - 1 α x e 3 x - 1 = β ⇒ lim x → 0 α x + 1 - e 3 x α x e 3 x - 1 3 x · 3 x = β ⇒ lim x → 0 α x + 1 - e 3 x 3 α x 2 = β as lim x → 0 e x - 1 x = 1 ⇒ lim x → 0 α - 3 e 3 x 6 α x = β [Using L' Hospital rule as 0 0   form ] Now for limit to exist, α = 3 So, lim x → 0 3 - 3 e 3 x 6 × 3 × x = β So, β = lim x → 0 - e 3 x - 1 6 x = lim x &

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