The function $f$ is defined piecewise as follows. \[ f(x)=\left\{\begin{array}{ll} x^{2}+3 x+2 & \text { if } x<-3 \\ (x+2)^{2}+2 & \text { if } x \geq-3 \end{array}\right. \] Find the following limits. If a limit does not exist, click on "Does Not Exist." (a) $\lim f(x)=$ \[ x \rightarrow-3^{-} \] (b) $\lim _{-3^{-}} f(x)=$ (c) $\lim _{x \rightarrow-3} f(x)=$

Step 1 ：The function $f$ is defined piecewise as follows: $f(x)=\left\{\begin{array}{ll} x^{2}+3 x+2 & \text { if } x<-3 \\ (x+2)^{2}+2 & \text { if } x \geq-3 \end{array}\right.$

Step 2 ：We are asked to find the limit of the function $f(x)$ as $x$ approaches $-3$ from the left, from the right, and in general.

Step 3 ：To find these limits, we substitute $x = -3$ into the appropriate piece of the piecewise function.

Step 4 ：For $x \rightarrow -3^{-}$, we use the first piece of the function, $x^{2}+3 x+2$, because this piece is defined for $x < -3$.

Step 5 ：For $x \rightarrow -3^{+}$ and $x \rightarrow -3$, we use the second piece of the function, $(x+2)^{2}+2$, because this piece is defined for $x \geq -3$.

Step 6 ：The limit of the function as $x$ approaches $-3$ from the left is $2$, and the limit as $x$ approaches $-3$ from the right is $3$.

Step 7 ：Since the left-hand limit and the right-hand limit are not equal, the limit of the function as $x$ approaches $-3$ does not exist.

Step 8 ：Final Answer: (a) $\lim_{x \rightarrow -3^{-}} f(x) = \boxed{2}$ (b) $\lim_{x \rightarrow -3^{+}} f(x) = \boxed{3}$ (c) $\lim_{x \rightarrow -3} f(x)$ does not exist.