If lim x → 0 α e x + β e - x + γ sin x x sin 2 x = 2 3 , where α , β , γ ∈ R , then which of the following is NOT correct?

Step-by-step solution

Given, lim x → 0 α e x + β e - x + γ sin x x sin 2 x = 2 3 ⇒ lim x → 0 α e x + β e - x + γ sin x x 3 sin 2 x x 2 = 2 3 ⇒ lim x → 0 α e x + β e - x + γ sin x x 3 = 2 3 Now using expansion of given function in terms of x we get, = lim x → 0 α 1 + x + x 2 2 ! + x 3 3 ! + … + β 1 - x + x 2 2 ! - x 3 3 ! + … + γ x - x 3 3 ! + … x 3 Now for limit to exist constant terms should be zero ⇒ a + β = 0 &