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_______.
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_______.
Correct Answer: 11
Solution and Explanation
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
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Concepts Used:
Limits And Derivatives
Mathematically, a limit is explained as a value that a function approaches as the input, and it produces some value. Limits are essential in calculus and mathematical analysis and are used to define derivatives, integrals, and continuity.
Limits Formula:
Derivatives of a Function:
A derivative is referred to the instantaneous rate of change of a quantity with response to the other. It helps to look into the moment-by-moment nature of an amount. The derivative of a function is shown in the below-given formula.
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