Is \( f(x)=\frac{x^{2}-11 x+18}{x-2} \) differentiable at \( x=2 \) ? Justify your answer. Choose the correct answer below. A. Yes, the function is differentiable because \( f(x) \) is continuous at \( x=2 \). B. No, the function is not differentiable because \( f(x) \) is not continuous at \( x=2 \). C. No, the function is not differentiable because \( f^{\prime}(x)=0 \) at \( x=2 \). D. Yes, the function is differentiable because \( f(x) \) is not continuous at \( x=2 \).

The given function is \[ f(x)=\frac{x^{2}-11x+18}{x-2}. \] **Step 1. Factor the numerator** Factor the quadratic expression in the numerator: \[ x^{2}-11x+18 = (x-2)(x-9). \] **Step 2. Simplify the expression for \( x\neq 2 \)** For \( x\neq 2 \), we can cancel the common factor \( (x-2) \): \[ f(x)=\frac{(x-2)(x-9)}{x-2} = x-9. \] So for all \( x\neq 2 \), the function simplifies to the linear function \( f(x)=x-9 \). **Step 3. Check the domain and continuity** Even though the expression simplifies to \( x-9 \) for \( x\neq 2 \), notice that the original function was defined as \[ f(x)=\frac{x^{2}-11x+18}{x-2}. \] Since the denominator \( x-2 \) is zero when \( x=2 \), the function \( f(x) \) is not defined at \( x=2 \). A necessary condition for differentiability at a point is that the function must be defined and continuous at that point. **Step 4. Conclusion on differentiability** Since \( f(x) \) is not defined at \( x=2 \), it is not continuous at \( x=2 \). Therefore, the function is not differentiable at \( x=2 \). The correct answer is: B. No, the function is not differentiable because \( f(x) \) is not continuous at \( x=2 \).