Let \( f(x)=\left\{\begin{array}{ll}10-x-x^{2} & \text { if } x \leq 2 \\ 2 x-7 & \text { if } x>2\end{array}\right. \) Calculate the following limits. Enter "DNE" if the limit does not exist. \( \lim _{x \rightarrow 2^{-}} f(x)=\square \) \( \lim _{x \rightarrow 2^{+}} f(x)=\square \) \( \lim _{x \rightarrow 2} f(x)=\square \)

To calculate the limits for the function \( f(x) \), we will evaluate the left-hand limit, right-hand limit, and the overall limit at \( x = 2 \). 1. **Left-hand limit** \( \lim _{x \rightarrow 2^{-}} f(x) \): Since we're approaching from the left, we use the first piece of the function \( f(x) = 10 - x - x^2 \): \[ \lim _{x \rightarrow 2^{-}} f(x) = 10 - 2 - 2^2 = 10 - 2 - 4 = 4 \] 2. **Right-hand limit** \( \lim _{x \rightarrow 2^{+}} f(x) \): As we approach from the right, we use the second piece of the function \( f(x) = 2x - 7 \): \[ \lim _{x \rightarrow 2^{+}} f(x) = 2(2) - 7 = 4 - 7 = -3 \] 3. **Overall limit** \( \lim _{x \rightarrow 2} f(x) \): Since the left-hand limit and the right-hand limit are not equal, the overall limit does not exist: \[ \lim _{x \rightarrow 2} f(x) = DNE \] Putting these results together, we have: \[ \lim _{x \rightarrow 2^{-}} f(x) = 4, \quad \lim _{x \rightarrow 2^{+}} f(x) = -3, \quad \lim _{x \rightarrow 2} f(x) = \text{DNE} \] So the final answers are: \[ \lim _{x \rightarrow 2^{-}} f(x) = 4 \] \[ \lim _{x \rightarrow 2^{+}} f(x) = -3 \] \[ \lim _{x \rightarrow 2} f(x) = DNE \]