Consider the following function, where \( n \) is a parameter. \( f(x)=\left\{\begin{array}{ll}2 n x+1 & \text { if } x<-1 \\ 7 & \text { if } x=-1 \\ -3 x+3 & \text { if } x>-1\end{array}\right. \) \( \lim _{x \rightarrow-1^{-}} f(x)=\square \) \( \lim _{x \rightarrow-1^{+}} f(x)= \) \( f(-1)= \) Submit Part

To analyze the function \( f(x) \) and find the limits and the value at \( x = -1 \), we will evaluate the left-hand limit, the right-hand limit, and the function value at \( x = -1 \). 1. **Left-hand limit as \( x \) approaches -1**: \[ \lim_{x \rightarrow -1^{-}} f(x) = \lim_{x \rightarrow -1^{-}} (2nx + 1) \] Since \( x < -1 \) in this case, we use the first piece of the piecewise function: \[ = 2n(-1) + 1 = -2n + 1 \] 2. **Right-hand limit as \( x \) approaches -1**: \[ \lim_{x \rightarrow -1^{+}} f(x) = \lim_{x \rightarrow -1^{+}} (-3x + 3) \] Since \( x > -1 \) in this case, we use the third piece of the piecewise function: \[ = -3(-1) + 3 = 3 + 3 = 6 \] 3. **Value of the function at \( x = -1 \)**: \[ f(-1) = 7 \] This is given directly from the second piece of the piecewise function. Now we can summarize the results: - \( \lim_{x \rightarrow -1^{-}} f(x) = -2n + 1 \) - \( \lim_{x \rightarrow -1^{+}} f(x) = 6 \) - \( f(-1) = 7 \) Thus, the answers are: - \( \lim_{x \rightarrow -1^{-}} f(x) = -2n + 1 \) - \( \lim_{x \rightarrow -1^{+}} f(x) = 6 \) - \( f(-1) = 7 \)