If $$\lim_{x \to \infty} \left[ \frac{x^2 + 1}{x + 1} - ax - b \right] = 0$$, then (a,b) is ​

Correct option is C

​$$\text{Given: } \lim_{x \to \infty} \left[ \frac{x+1}{x^2 + 1} - ax - b \right] = 0 \\[8pt]\text{Step 1: Polynomial division} \\\frac{x+1}{x^2 + 1} = x - 1 + \frac{2}{x+1} \\[6pt]\text{So, } \lim_{x \to \infty} \left[ x - 1 + \frac{2}{x+1} - ax - b \right] = 0 \\[6pt]\text{Group terms: } \\= (1 - a)x + (-1 - b) + \frac{2}{x+1} \\[6pt]\text{For the limit to be 0 as } x \to \infty: \\1 - a = 0 \Rightarrow a = 1 \\-1 - b = 0 \Rightarrow b = -1 \\[6pt]\text{Final Result: } \boxed{(a, b) = (1, -1)}$$​​