Problem Analysis #17

[THOUGHT] The limit condition requires that the expression inside the limit has a finite value as (x \to 0). Expanding the integral and the cosine term, we find that the coefficient of (1/x^2) must be zero to avoid divergence, leading to (\frac{a}{2} + b = 0). The constant term then gives (\frac{a}{6} - \frac{b}{2} = 2). Solving these equations yields (a = \frac{24}{5}) and (b = -\frac{12}{5}). Thus, (a + b = \frac{12}{5}), ((a + b)^2 = \frac{144}{25}), and (100(a + b)^2 = 576). [/THOUGHT]

\boxed{576}

[THOUGHT] We need to solve for a and b using the given equations. We will use Python to solve the system of equations. [/THOUGHT] [PYTHON] from sympy import symbols, Eq, solve

a, b = symbols('a b') eq1 = Eq(b, -a/2) eq2 = Eq(a/6 - b/2, 2)

solution = solve((eq1, eq2), (a, b)) print(solution) [/PYTHON]

\boxed{576}