Question

If \(\underset{x\rightarrow 1}{lim}\frac{sin(3x^2-4x+1)-x^2+1}{2x^3-7x^2+ax+b} = -2\), then the quadratic equation having roots \(a\) and \(b\) is

• A. \(x^2+5x-24 = 0\)
• B. \(x^2-5x+24 = 0\)
• C. \(x^2-5x-24 = 0\)
• D. \(x^2+5x+24 = 0\)

Hint:

The denominator must vanish at \(x=1\) twice for the limit to be finite and nonzero.

Answer 1

The Correct Option is C

Step 1: Find the two conditions
Finite limit forces $a+b=5$. A limit of $-2$ rather than $0$ needs the first derivative of the denominator to vanish too: $a=8$.

Step 2: Verify and build
With $a=8,b=-3$ the second derivative ratio is $4/(-2)=-2$, as given. The quadratic with roots $8,-3$ is $x^2-(8-3)x+(8)(-3)=x^2-5x-24=0$, option (C).

Final Answer:
The quadratic is $x^2-5x-24=0$, option (C). \[ \boxed{x^2-5x-24=0} \]