Question:

Step 1: Perform polynomial long division on the rational function. We need to divide $x^3 + 1$ by $x^2 + 1$ to simplify the expression. Using polynomial long division: $$x^3 + 1 = x(x^2 + 1) - x + 1$$ Therefore: $$\frac{x^3+1}{x^2+1} = x + \frac{-x+1}{x^2+1}$$ Step 2: Rewrite the limit expression using the simplified form. Substitute the simplified form into the original limit expression: $$\frac{x^3+1}{x^2+1} - (ax+b) = x + \frac{-x+1}{x^2+1} - ax - b$$ Combine like terms: $$= (1-a)x - b + \frac{-x+1}{x^2+1}$$ Step 3: Determine the condition for the limit to be finite. For the limit to exist and be finite as $x \to \infty$, the coefficient of $x$ in the expression must equal zero. Otherwise, the expression would grow unboundedly. Therefore: $$1 - a = 0 \implies a = 1$$ Step 4: Evaluate the limit with the value of $a$. With $a = 1$, the expression becomes: $$-b + \frac{-x+1}{x^2+1}$$ As $x \to \infty$, the fraction $\frac{-x+1}{x^2+1} \to 0$ (since the denominator grows faster than the numerator). Therefore: $$\lim_{x \to \infty} \left[\frac{x^3+1}{x^2+1} - (ax+b)\right] = -b + 0 = -b$$ Step 5: Solve for $b$ using the given limit value. We are given that this limit equals $2$: $$-b = 2 \implies b = -2$$ **Final Answer:** $a = 1$ and $b = -2$ This corresponds to **Option 3**.