If lim _(x → 1) (sin (3x² - 4x + 1) - x² + 1)/(2x³ - 7x² + ax + b) = - 2, then the value of (a - b) is equal to ___________.

Solution

Maths sir · 4 steps

The numerator vanishes twice at x = 1, so the denominator must vanish twice too. That gives a and b.

1. First condition. At x = 1 the numerator is sin 0 - 1 + 1 = 0. For a finite limit the denominator must also be 0 there: 2 - 7 + a + b = 0, so a + b = 5.
2. Differentiate once. By L'Hôpital's rule, the numerator's derivative is cos(3x² - 4x + 1)(6x - 4) - 2x. At x = 1 this is 1 × 2 - 2 = 0, so the numerator is still 0. The limit can only be the finite value -2 if the denominator's derivative 6x² - 14x + a is also 0 at x = 1. That gives 6 - 14 + a = 0, so a = 8 and b = 5 - 8 = -3.
3. Check the limit. Differentiate again and put x = 1. The numerator gives -sin(0)(2)² + 6 cos 0 - 2 = 4, and the denominator gives 12 - 14 = -2.

   lim(x → 1) (sin(3x² - 4x + 1) - x² + 1)/(x-1)²(2x-3) = 4/-2 = -2

4. So. The limit is -2 as required, so a = 8, b = -3 and a - b = 8 - (-3) = 11.

The common slip: 5. Writing b = 8 - 5 = 3 instead of b = 5 - 8 = -3 gives a - b = 5, not 11.

The idea to keep. If the numerator has a double zero at the limit point, the denominator needs one too for a finite nonzero limit. Each zero gives you one equation in the unknowns.

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