Question

If
$\lim_{{x \to 1}} \frac{{\sin(3x^2 - 4x + 1) - x^2 + 1}}{{2x^3 - 7x^2 + ax + b}} = -2$
, then the value of (a – b) is equal to_______.

Answer 1

The correct answer is 11
$\lim_{{x \to 1}} \frac{({\frac{\sin(3x^2 - 4x + 1)}{3x^2 - 4x + 1}) \cdot (3x^2 - 4x + 1) - (x^2 + 1)}}{{2x^3 - 7x^2 + ax + b}} = -2$
$\lim_{{x \to 1}} \frac{{3x^2 - 4x + 1 - x^2 + 1}}{{2x^3 - 7x^2 + ax + b}} = -2$
$\lim_{{x \to 1}} \frac{{2(x-1)^2}}{{2x^3 - 7x^2 + ax + b}} = -2$
So f(x) = 2x3 – 7x2 + ax +b = 0 has x = 1 as repeated root, therefore f(1) = 0 and f ′(1) = 0 gives
a + b + 5 and a = 8
So, a – b = 11