Graph f, locate all points of disconlinuity, and discuss the behavior of fat these points. \[ f(x)=\left\{\begin{array}{ll} 1+x & \text { if } x<1 \\ 6-x & \text { if } x \geq 1 \end{array}\right. \] Discuss the behavior off at its point(s) of discontinuity. Find \( \mathrm{f}(\mathrm{x}) \) at any points of discontinuily Select the correct choico bolow and, if nocessary, fill in the answer box to completo your choice. A. At the point(s) of discontinuity, the value(s) of the function \( f(x) \) is/are \( \square \) (Use a comma lo separate answers as needed.) B. There are no points of discontinuity. Find \( \lim _{x \rightarrow c} f(x) \). where \( c \) is/are the poinl( \( (s) \) at which \( f(x) \) is dscontinuous. Select the correct choice below and, if necessary, fill in the answer box lo ccmplete your choice. A. \( \lim _{x \rightarrow c} f(x)= \) \( \square \) (Use a cormma to saparale answers as needed.) B. The limit does not exist

Let's analyze the given piecewise function to identify any points of discontinuity and discuss the behavior at those points. \[ f(x) = \begin{cases} 1 + x & \text{if } x < 1 \\ 6 - x & \text{if } x \geq 1 \end{cases} \] ### 1. **Graphing the Function** - **For \(x < 1\):** The function \(f(x) = 1 + x\) is a straight line with a slope of 1 and a y-intercept at \((0, 1)\). - **For \(x \geq 1\):** The function \(f(x) = 6 - x\) is a straight line with a slope of -1 and a y-intercept at \((0, 6)\). ### 2. **Identifying Points of Discontinuity** The potential point of discontinuity occurs where the definition of the function changes, which is at \(x = 1\). - **Left-Hand Limit as \(x \to 1^-\):** \[ \lim_{x \to 1^-} f(x) = 1 + 1 = 2 \] - **Right-Hand Limit as \(x \to 1^+\):** \[ \lim_{x \to 1^+} f(x) = 6 - 1 = 5 \] - **Function Value at \(x = 1\):** \[ f(1) = 6 - 1 = 5 \] Since \(\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x)\), the function has a **jump discontinuity** at \(x = 1\). ### 3. **Behavior at the Point of Discontinuity** - **Value of the Function at Discontinuity:** \(f(1) = 5\) - **Limit at the Point of Discontinuity:** The limit \(\lim_{x \to 1} f(x)\) **does not exist** because the left-hand and right-hand limits are not equal. ### 4. **Multiple Choice Selections** **A. At the point(s) of discontinuity, the value(s) of the function \( f(x) \) is/are:** \[ \boxed{5} \] **B. There are no points of discontinuity.** \[ \boxed{\text{Incorrect}} \] **Limit at the Point of Discontinuity:** **A. \( \lim _{x \rightarrow 1} f(x)= \)** \[ \boxed{\text{Not Applicable}} \] **B. The limit does not exist** \[ \boxed{\text{Correct}} \] ### 5. **Summary** - **Point of Discontinuity:** \(x = 1\) - **Value at Discontinuity:** \(f(1) = 5\) - **Limit at Discontinuity:** Does not exist