Let \( f(x)=\left\{\begin{array}{ll}2 x+9 & \text { if } x<-3 \\ \sqrt{x+12} & \text { if } x>-3 \\ 2 & \text { if } x=-3\end{array}\right. \) Select all statements below that you agree with. Note: You may be checking more than one box. No partial credit. \( \square f(-3) \) is defined. \( \lim _{x \rightarrow-3} f(x) \) exists. \( \lim _{x \rightarrow-3} f(x)=f(-3) \). \( \square \) The function is continuous at \( x=-3 \). \( \square \) The function is not continuous at \( x=-3 \).

Let's analyze the function \( f(x) \) and the statements provided. 1. **Check if \( f(-3) \) is defined:** \[ f(-3) = 2 \quad \text{(since the definition states that if } x = -3, f(x) = 2\text{)} \] Therefore, \( f(-3) \) is defined. 2. **Check if \( \lim_{x \rightarrow -3} f(x) \) exists:** We need to find the left-hand limit and the right-hand limit as \( x \) approaches \(-3\). - **Left-hand limit** (\( x \to -3^- \)): \[ f(x) = 2x + 9 \quad \text{for } x < -3 \] \[ \lim_{x \to -3^-} f(x) = 2(-3) + 9 = -6 + 9 = 3 \] - **Right-hand limit** (\( x \to -3^+ \)): \[ f(x) = \sqrt{x + 12} \quad \text{for } x > -3 \] \[ \lim_{x \to -3^+} f(x) = \sqrt{-3 + 12} = \sqrt{9} = 3 \] Since both the left-hand limit and right-hand limit are equal: \[ \lim_{x \to -3} f(x) = 3 \] Thus, \( \lim_{x \rightarrow -3} f(x) \) exists. 3. **Check if \( \lim_{x \rightarrow -3} f(x) = f(-3) \):** We found that: \[ \lim_{x \to -3} f(x) = 3 \quad \text{and} \quad f(-3) = 2 \] Since \( 3 \neq 2 \), we have: \[ \lim_{x \rightarrow -3} f(x) \neq f(-3) \] 4. **Check if the function is continuous at \( x = -3 \):** A function is continuous at a point if: \[ \lim_{x \to c} f(x) = f(c) \] Since we found that \( \lim_{x \to -3} f(x) = 3 \) and \( f(-3) = 2 \), the function is not continuous at \( x = -3 \). Now, let's summarize the statements: - \( \square f(-3) \) is defined. **(True)** - \( \lim_{x \rightarrow -3} f(x) \) exists. **(True)** - \( \lim_{x \rightarrow -3} f(x) = f(-3) \). **(False)** - \( \square \) The function is continuous at \( x = -3 \). **(False)** - \( \square \) The function is not continuous at \( x = -3 \). **(True)** Thus, the correct selections are: - \( \square f(-3) \) is defined. - \( \lim_{x \rightarrow -3} f(x) \) exists. - \( \square \) The function is not continuous at \( x = -3 \).