Find the values of a and b if lim (x->inf) [ (x3+1)/(x2+1) - (ax+b) ] = 2.

Alright Anantaram — let's do this. To find the values of a and b, we first simplify the expression inside the limit. Step 1: Combine the terms into a single fraction. _x ( (x^3+1)/(x^2+1) - (ax+b) ) = 2 _x ( (x^3+1 - (ax+b)(x^2+1))/(x^2+1) ) = 2 Expand the product (ax+b)(x^2+1): (ax+b)(x^2+1) = ax(x^2+1) + b(x^2+1) = ax^3 + ax + bx^2 + b Substitute this back into the numerator: x^3+1 - (ax^3 + bx^2 + ax + b) = x^3+1 - ax^3 - bx^2 - ax - b Rearrange the terms by powers of x: (1-a)x^3 - bx^2 - ax + (1-b) So the limit becomes: _x ( ((1-a)x^3 - bx^2 - ax + (1-b))/(x^2+1) ) = 2 Step 2: Determine the value of a. For the limit of a rational function as x to be a finite non-zero value, the degree of the numerator must be equal to the degree of the denominator. The degree of the denominator is 2. The current degree of the numerator is 3 (due to the x^3 term). For the limit to be finite, the coefficient of x^3 in the numerator must be zero. 1-a = 0 a = 1 Step 3: Substitute a=1 into the expression and evaluate the limit. With a=1, the expression becomes: _x ( ((1-1)x^3 - bx^2 - 1x + (1-b))/(x^2+1) ) = _x ( (-bx^2 - x + (1-b))/(x^2+1) ) To evaluate this limit, divide every term in the numerator and denominator by the highest power of x in the denominator, which is x^2: _x ( (-bx^2)/(x^2) - (x)/(x^2) + (1-b)/(x^2)(x^2)/(x^2) + (1)/(x^2) ) = _x ( (-b - 1)/(x) + (1-b)/(x^2)1 + (1)/(x^2) ) As x , the terms (1)/(x) and (1)/(x^2) approach 0. (-b - 0 + 0)/(1 + 0) = -b Step 4: Set the result of the limit equal to 2 and solve for b. -b = 2 b = -2 Step 5: Compare the values of a and b with the given options. We found a=1 and b=-2. This matches option (C). The final answer is C) a = 1, b = -2. Send me the next one 📸