Let f(x) = x⁶ + 2x⁴ + x³ + 2x + 3, x ∈ R. Then the natural number n for which lim _(x → 1) (xⁿf(1) - f(x))/(x - 1) = 44 is __________.

Solution

Maths sir · 4 steps

Put x = 1 first: you get 0/0, so differentiate the top and bottom and the limit becomes 9n - f'(1).

1. Check the form. f(1) = 1 + 2 + 1 + 2 + 3 = 9. At x = 1 the numerator is 1ⁿ × 9 - 9 = 0 and the denominator is 0, so the limit has the 0/0 form and L'Hopital's rule applies.
2. Differentiate. Remember that f(1) = 9 is a constant. The derivative of the numerator is 9n xⁿ⁻¹ - f'(x), and the derivative of the denominator is 1.

   lim(x → 1) (xⁿ f(1) - f(x))/(x - 1) = 9n - f'(1)

3. Find f'(1). f'(x) = 6x⁵ + 8x³ + 3x² + 2, so f'(1) = 6 + 8 + 3 + 2 = 19.
4. So. 9n - 19 = 44 gives 9n = 63, so n = 7.

The idea to keep. When a limit gives 0/0 at x = a, differentiate the top and bottom. Values like f(1) are plain numbers, so treat them as constants while differentiating.

Answer from the NTA final answer key. Solution by PrepWiser.