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NTA Abhyas JEE Main2020MathematicsLimitsPractice

If l i m x → 0 x 1 + a cos ⁡ x - b sin ⁡ x x 3 = 1 , then the value of a b is equal to

Correct answer

3.75

Step-by-step solution

Using the expansion, we get, l i m x → 0 x + a x 1 - x 2 2 ! + x 4 4 ! - . . . . . . - b x - x 3 3 ! + . . . . . . x 3 = 1 l i m x → 0 1 + a - b x + - a 2 + b 6 x 3 + . . . . . x 3 = 1 For limit to exist numerator & denominator must be of the same degree ∴ 1 + a - b = 0 … 1 Also, - a 2 + b 6 = 1 ⇒ 3 a - b + 6 = 0 … 2 By equations 1 & 2 , we get, a = - 5 2 and b = - 3 2 ⇒ a b = 15 4 = 3 .75

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