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

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**Step 1. Identify the key features from the limits and function values.**

- At $$ x=3 $$:
  - The left‐hand limit is $$\displaystyle \lim_{x	o3^-}f(x)=2$$.
  - The right‐hand limit is $$\displaystyle \lim_{x	o3^+}f(x)=4$$.
  - 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 $$\lim_{xightarrow -2^{2}} f(x)=2$$ is understood here as the limit of $$f(x)$$ as $$ x $$ approaches $$-2$$ is $$2$$; in other words, $$\lim_{x	o-2}f(x)=2$$.
  - 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 $$(-2,1)$$.

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**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 $$\lim_{x	o-2}f(x)=2$$) 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.

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**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 $$(3,3)$$.

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**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 $$\lim_{x	o3^-}f(x)=2$$.
  - Mark an open circle at $$(3,4)$$ indicating $$\lim_{x	o3^+}f(x)=4$$.
  - 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.
To sketch the graph: 1. **At $$ x = -2 $$:** - Open circle at $$(-2, 2)$$ (limit from both sides). - Solid dot at $$(-2, 1)$$ (function value). 2. **At $$ x = 3 $$:** - Open circle at $$(3, 2)$$ (left limit). - Open circle at $$(3, 4)$$ (right limit). - Solid dot at $$(3, 3)$$ (function value). 3. **Between $$ x = -2 $$ and $$ x = 3 $$:** - Draw a smooth curve approaching $$(-2, 2)$$ from the left. - The curve should approach $$(3, 2)$$ from the left and $$(3, 4)$$ from the right. 4. **Beyond $$ x = 3 $$:** - Continue the curve based on the function's behavior. Label the $$x$$-axis with $$-2$$ and $$3$$, and the $$y$$-axis with $$1$$, $$2$$, $$3$$, and $$4$$.

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