\( F(x)=\left\{\begin{array}{ll}3 x-6, & \text { for } x<1 \\ x, & \text { for } x \geq 1\end{array}\right. \); Find \( \lim _{x \rightarrow 1^{-}} F(x), \lim _{x \rightarrow 1^{+}} F(x), \lim _{x \rightarrow 1} F(x) \)

To find the limits of the function \( F(x) \) as \( x \) approaches 1 from the left and from the right, we will evaluate the function based on its piecewise definition. 1. **Finding \( \lim_{x \rightarrow 1^{-}} F(x) \)**: For \( x < 1 \), the function is defined as \( F(x) = 3x - 6 \). We will evaluate this limit as \( x \) approaches 1 from the left: \[ \lim_{x \rightarrow 1^{-}} F(x) = \lim_{x \rightarrow 1^{-}} (3x - 6) = 3(1) - 6 = 3 - 6 = -3. \] 2. **Finding \( \lim_{x \rightarrow 1^{+}} F(x) \)**: For \( x \geq 1 \), the function is defined as \( F(x) = x \). We will evaluate this limit as \( x \) approaches 1 from the right: \[ \lim_{x \rightarrow 1^{+}} F(x) = \lim_{x \rightarrow 1^{+}} x = 1. \] 3. **Finding \( \lim_{x \rightarrow 1} F(x) \)**: The overall limit \( \lim_{x \rightarrow 1} F(x) \) exists if both one-sided limits are equal. Since we found: \[ \lim_{x \rightarrow 1^{-}} F(x) = -3 \quad \text{and} \quad \lim_{x \rightarrow 1^{+}} F(x) = 1, \] these two limits are not equal. Therefore, the limit as \( x \) approaches 1 does not exist: \[ \lim_{x \rightarrow 1} F(x) \text{ does not exist.} \] In summary, we have: - \( \lim_{x \rightarrow 1^{-}} F(x) = -3 \) - \( \lim_{x \rightarrow 1^{+}} F(x) = 1 \) - \( \lim_{x \rightarrow 1} F(x) \) does not exist.