Consider the following function. \[ f(x)=\left\{\begin{array}{lll}x^{2}+x & \text { if } \quad x<-1 \\ 0 & \text { if } \quad x=-1 \\ 2 x-1 & \text { if } \quad x>-1\end{array}\right. \] Evaluate the given limits. If a limit does not exist, enter DNE. (a) \( \lim _{x \rightarrow-1^{-}} f(x)=\square \) (b) \( \lim _{x \rightarrow-1^{+}} f(x)=\square \) (c) \( \lim _{x \rightarrow-1} f(x)=\square \)

To evaluate the limits for the function \( f(x) \), we will analyze the behavior of \( f(x) \) as \( x \) approaches \(-1\) from the left and from the right. The function is defined as follows: \[ f(x)=\left\{ \begin{array}{lll} x^{2}+x & \text { if } \quad x<-1 \\ 0 & \text { if } \quad x=-1 \\ 2 x-1 & \text { if } \quad x>-1 \end{array} \right. \] ### (a) \( \lim _{x \rightarrow-1^{-}} f(x) \) To find the left-hand limit as \( x \) approaches \(-1\), we use the piece of the function defined for \( x < -1 \): \[ f(x) = x^2 + x \] Now, we compute the limit: \[ \lim_{x \to -1^{-}} f(x) = \lim_{x \to -1^{-}} (x^2 + x) \] Substituting \( x = -1 \): \[ = (-1)^2 + (-1) = 1 - 1 = 0 \] Thus, \[ \lim _{x \rightarrow-1^{-}} f(x) = 0 \] ### (b) \( \lim _{x \rightarrow-1^{+}} f(x) \) For the right-hand limit as \( x \) approaches \(-1\), we use the piece of the function defined for \( x > -1 \): \[ f(x) = 2x - 1 \] Now, we compute the limit: \[ \lim_{x \to -1^{+}} f(x) = \lim_{x \to -1^{+}} (2x - 1) \] Substituting \( x = -1 \): \[ = 2(-1) - 1 = -2 - 1 = -3 \] Thus, \[ \lim _{x \rightarrow-1^{+}} f(x) = -3 \] ### (c) \( \lim _{x \rightarrow-1} f(x) \) The overall limit as \( x \) approaches \(-1\) exists only if the left-hand limit and the right-hand limit are equal. We found: \[ \lim _{x \rightarrow-1^{-}} f(x) = 0 \] \[ \lim _{x \rightarrow-1^{+}} f(x) = -3 \] Since these two limits are not equal, the limit does not exist: \[ \lim _{x \rightarrow-1} f(x) = \text{DNE} \] ### Final Answers: (a) \( \lim _{x \rightarrow-1^{-}} f(x) = 0 \) (b) \( \lim _{x \rightarrow-1^{+}} f(x) = -3 \) (c) \( \lim _{x \rightarrow-1} f(x) = \text{DNE} \)