Sketch a graph of the following function that satisfies all the conditions. Make sure you label your axis. Show work in WRITTEN WORK! \[ \begin{array}{l}\lim _{x \rightarrow 3^{+}} f(x)=4, \lim _{x \rightarrow 3^{-}} f(x)=2, \quad \lim _{x \rightarrow-2^{2}} f(x)=2, \\ f(3)=3, \quad f(-2)=1\end{array} \]

**Step 1. Identify the key features from the limits and function values.** - At \( x=3 \): - The left‐hand limit is \(\displaystyle \lim_{x\to3^-}f(x)=2\). - The right‐hand limit is \(\displaystyle \lim_{x\to3^+}f(x)=4\). - The actual function value is \(f(3)=3\). This tells us that as \( x \) approaches 3 from the left the graph approaches the point \((3,2)\) (with an open circle there) and as \( x \) approaches 3 from the right the graph approaches the point \((3,4)\) (again an open circle). However, the graph is defined at \( x=3 \) by the point \((3,3)\) (a solid dot). - At \( x=-2 \): - The notation \(\lim_{x\rightarrow -2^{2}} f(x)=2\) is understood here as the limit of \(f(x)\) as \( x \) approaches \(-2\) is \(2\); in other words, \(\lim_{x\to-2}f(x)=2\). - The function value is \(f(-2)=1\). Thus, as \(x\) nears \(-2\) (from both sides) the value of \(f(x)\) tends to \(2\) (open circle at \((-2,2)\)), but the point on the graph at \(x=-2\) is given by the solid dot at \((-2,1)\). --- **Step 2. Sketching the Graph Piecewise** 1. **For \(x<-2\) and in an interval around \(-2\):** We can imagine that the graph is drawn as a continuous curve except at \(x=-2\). - As the graph nears \(x=-2\) from either side, it gets closer to \(y=2\). - At \(x=-2\) there is a hole at the point \((-2,2)\) (because \(\lim_{x\to-2}f(x)=2\)) and a filled dot at \((-2,1)\) (because \(f(-2)=1\)). 2. **For values of \(x\) near 3:** Again, assume the function is drawn continuously everywhere else except at \(x=3\). - As \(x\) approaches 3 from the left, the curve approaches the point \((3,2)\); mark an open circle at \((3,2)\). - As \(x\) approaches 3 from the right, the curve approaches the point \((3,4)\); mark an open circle at \((3,4)\). - At \(x=3\) the function is defined to be \(3\), so plot a solid dot at \((3,3)\). 3. **Connecting the Pieces:** You can join the parts of the graph with smooth curves (or linear segments, if preferred) ensuring that: - The part of the graph coming from the left of \(x=-2\) smoothly approaches \((-2,2)\) even though the function is defined to have a different value at \(x=-2\). - Similarly, around \(x=3\) the parts of the graph on the left and right approach their respective limits (2 and 4) with a jump discontinuity, while a solid point is placed at \((3,3)\) for the function’s defined value. --- **Step 3. Labeling the Axes and Annotating Key Points** - Draw the horizontal line (the \(x\)-axis) and the vertical line (the \(y\)-axis); label them accordingly. - Mark the key \(x\)-values, in particular \(x=-2\) and \(x=3\). - On the \(y\)-axis, label the values \(1\), \(2\), \(3\), and \(4\) (since these are the key values in the problem). - At \(x=-2\): - Mark an open circle at \((-2,2)\). - Mark a filled dot at \((-2,1)\). - At \(x=3\): - Mark an open circle at \((3,2)\) (coming from the left). - Mark an open circle at \((3,4)\) (coming from the right). - Mark a filled dot at \((3,3)\). --- **Step 4. Final Graph Description** - **Left of \(x=-2\):** Sketch a smooth curve that approaches the point \((-2,2)\) as \(x\) nears \(-2\) from both sides. - **At \(x=-2\):** - Put an open circle at \((-2,2)\). - Plot a solid dot at \((-2,1)\) to show the defined value. - **Between \(x=-2\) and \(x=3\):** Have the graph continue (in a manner of your choice, e.g., a smooth curve or a piecewise linear function) but arranged so that as \(x\) approaches \(3\) from the left the curve goes toward \((3,2)\). - **At \(x=3\):** - Mark an open circle at \((3,2)\) indicating \(\lim_{x\to3^-}f(x)=2\). - Mark an open circle at \((3,4)\) indicating \(\lim_{x\to3^+}f(x)=4\). - Plot the point \((3,3)\) as a solid dot to indicate \(f(3)=3\). - **Right of \(x=3\):** Sketch the graph such that as \(x\) increases just past 3, the curve goes toward \((3,4)\) and continues further from that value. This graph reflects all the given conditions with properly labeled axes.