Determine whether the following function is continuous at a. Use the continuity checklist to justify your answer \( f(x)=\frac{2 x^{2}+5 x+3}{x^{2}+8 x} a=-8 \) \( \square \) A. The function is continuous at \( a=-8 \). \( \square \) B. The function is not continuous at \( a=-8 \) because \( f(-8) \) is undefined \( \square \) C. The function is not continuous at \( a=-8 \) because \( \lim _{x \rightarrow-8} f(x) \) does not exist. \( \square \) D. The function is not continuous at \( a=-8 \) because \( \lim _{x \rightarrow-8} f(x) \neq f(-8) \).

To determine whether the function \( f(x) = \frac{2x^2 + 5x + 3}{x^2 + 8x} \) is continuous at \( a = -8 \), we will use the continuity checklist, which involves checking three conditions: 1. **The function \( f(a) \) is defined.** 2. **The limit \( \lim_{x \to a} f(x) \) exists.** 3. **The limit equals the function value, i.e., \( \lim_{x \to a} f(x) = f(a) \).** Let's evaluate these conditions step by step. ### Step 1: Check if \( f(-8) \) is defined First, we need to find \( f(-8) \): \[ f(-8) = \frac{2(-8)^2 + 5(-8) + 3}{(-8)^2 + 8(-8)} \] Calculating the numerator: \[ 2(-8)^2 + 5(-8) + 3 = 2(64) - 40 + 3 = 128 - 40 + 3 = 91 \] Calculating the denominator: \[ (-8)^2 + 8(-8) = 64 - 64 = 0 \] Since the denominator is zero, \( f(-8) \) is undefined. ### Step 2: Check if \( \lim_{x \to -8} f(x) \) exists Next, we need to find the limit as \( x \) approaches -8: \[ \lim_{x \to -8} f(x) = \lim_{x \to -8} \frac{2x^2 + 5x + 3}{x^2 + 8x} \] We can factor the denominator: \[ x^2 + 8x = x(x + 8) \] As \( x \) approaches -8, the denominator approaches 0. We also need to check the numerator at \( x = -8 \): \[ 2(-8)^2 + 5(-8) + 3 = 91 \quad \text{(as calculated before)} \] Since the numerator approaches 91 and the denominator approaches 0, the limit does not exist (it approaches \( \frac{91}{0} \), which is undefined). ### Step 3: Conclusion Since \( f(-8) \) is undefined and the limit \( \lim_{x \to -8} f(x) \) does not exist, we can conclude that the function is not continuous at \( a = -8 \). Thus, the correct answer is: \( \square \) A. The function is continuous at \( a=-8 \). \( \square \) B. The function is not continuous at \( a=-8 \) because \( f(-8) \) is undefined. \( \square \) C. The function is not continuous at \( a=-8 \) because \( \lim _{x \rightarrow-8} f(x) \) does not exist. \( \square \) D. The function is not continuous at \( a=-8 \) because \( \lim _{x \rightarrow-8} f(x) \neq f(-8) \). The correct choices are **B**, **C**, and **D**.