Print first and last names of each group member at the top of the page. 1. Sketch the graph of a function $$ f $$ with all the following properties. Identify and label any points of discontinuity on the interval $$ (-\infty, \infty) $$ $$ \lim _{x \rightarrow-9^{-}} f(x)=\infty \quad \lim _{x \rightarrow-5} f(x)=\infty \quad \lim _{x \rightarrow 2^{+}} f(x)=-4 $$ $$ \lim _{x \rightarrow-9^{+}} f(x)=-\infty \quad \lim _{x \rightarrow 2^{-}} f(x)=1 \quad f(2)=5 $$

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**Group Members:**
- [First Name] [Last Name]
- [First Name] [Last Name]
- [First Name] [Last Name]

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### Problem 1

**Sketch the graph of a function $$ f $$ with all the following properties. Identify and label any points of discontinuity on the interval $$ (-\infty, \infty) $$:**

$$
\lim _{x \rightarrow -9^{-}} f(x) = \infty \quad \lim _{x \rightarrow -5} f(x) = \infty \quad \lim _{x \rightarrow 2^{+}} f(x) = -4
$$
$$
\lim _{x \rightarrow -9^{+}} f(x) = -\infty \quad \lim _{x \rightarrow 2^{-}} f(x) = 1 \quad f(2) = 5
$$

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#### **Solution:**

To sketch the graph of the function $$ f $$ with the given properties, we'll analyze each condition step by step and determine the behavior of $$ f $$ near the specified points.

1. **Behavior near $$ x = -9 $$:**
   - $$ \lim _{x \rightarrow -9^{-}} f(x) = \infty $$: As $$ x $$ approaches $$-9$$ from the left, $$ f(x) $$ tends to positive infinity.
   - $$ \lim _{x \rightarrow -9^{+}} f(x) = -\infty $$: As $$ x $$ approaches $$-9$$ from the right, $$ f(x) $$ tends to negative infinity.
   
   **Conclusion:** There is a vertical asymptote at $$ x = -9 $$, and the function has a jump discontinuity at this point.

2. **Behavior near $$ x = -5 $$:**
   - $$ \lim _{x \rightarrow -5} f(x) = \infty $$: As $$ x $$ approaches $$-5$$ from both sides, $$ f(x) $$ tends to positive infinity.
   
   **Conclusion:** There is a vertical asymptote at $$ x = -5 $$. Since both one-sided limits approach the same infinity, this is an infinite discontinuity (also known as a vertical asymptote).

3. **Behavior near $$ x = 2 $$:**
   - $$ \lim _{x \rightarrow 2^{-}} f(x) = 1 $$: As $$ x $$ approaches $$2$$ from the left, $$ f(x) $$ approaches $$1$$.
   - $$ \lim _{x \rightarrow 2^{+}} f(x) = -4 $$: As $$ x $$ approaches $$2$$ from the right, $$ f(x) $$ approaches $$-4$$.
   - $$ f(2) = 5 $$: The function is defined at $$ x = 2 $$ with a value of $$5$$.
   
   **Conclusion:** There is a removable discontinuity (a "hole") at $$ x = 2 $$ since the left and right limits exist but are not equal to each other or to $$ f(2) $$.

4. **Behavior elsewhere ($$ x $$ not equal to $$-9$$, $$-5$$, or $$2$$):**
   - To complete the sketch, we need to define the behavior of $$ f(x) $$ in other regions. Since specific behaviors are not provided, we can assume $$ f(x) $$ behaves smoothly elsewhere, possibly as a rational function that includes the identified vertical asymptotes and the behavior around $$ x = 2 $$.

#### **Sketching the Graph:**

While I cannot provide a visual sketch here, I can describe how to construct it:

- **Vertical Asymptote at $$ x = -9 $$:** Draw a dashed vertical line at $$ x = -9 $$. As $$ x $$ approaches $$-9$$ from the left, $$ f(x) $$ rises to positive infinity. From the right, it plunges to negative infinity.
  
- **Vertical Asymptote at $$ x = -5 $$:** Draw another dashed vertical line at $$ x = -5 $$. $$ f(x) $$ approaches positive infinity from both sides as $$ x $$ approaches $$-5$$.
  
- **Behavior at $$ x = 2 $$:** 
  - Plot a point at $$ (2, 5) $$.
  - Draw the graph approaching $$1$$ from the left of $$2$$ and $$-4$$ from the right of $$2$$. Indicate the discontinuity with an open circle at both $$ (2, 1) $$ and $$ (2, -4) $$ to show the limits, and a filled circle at $$ (2, 5) $$ to show the function value.
  
- **General Behavior:**
  - Between $$ x = -9 $$ and $$ x = -5 $$, and between $$ x = -5 $$ and $$ x = 2 $$, choose appropriate functions that align with the asymptotic behavior and limits.
  - For $$ x < -9 $$ and $$ x > 2 $$, define $$ f(x) $$ smoothly, ensuring continuity except at the specified points.

#### **Points of Discontinuity:**

1. **$$ x = -9 $$:** Infinite discontinuity (vertical asymptote) with different behaviors from the left and right.
2. **$$ x = -5 $$:** Infinite discontinuity (vertical asymptote) with $$ f(x) $$ approaching positive infinity from both sides.
3. **$$ x = 2 $$:** Removable discontinuity with $$ f(x) $$ defined as $$5$$ at this point, while the limits from the left and right do not equal each other or the function value.

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### Summary of Discontinuities:

- **Vertical Asymptotes:** $$ x = -9 $$ and $$ x = -5 $$.
- **Removable Discontinuity:** $$ x = 2 $$.

Ensure to label these points clearly on your graph when sketching.
**Group Members:** - [First Name] [Last Name] - [First Name] [Last Name] - [First Name] [Last Name] --- **Problem 1:** **Sketch the graph of a function $$ f $$ with the following properties:** - As $$ x $$ approaches $$-9$$ from the left, $$ f(x) $$ goes to positive infinity. - As $$ x $$ approaches $$-9$$ from the right, $$ f(x) $$ goes to negative infinity. - As $$ x $$ approaches $$-5$$, $$ f(x) $$ goes to positive infinity. - As $$ x $$ approaches $$2$$ from the left, $$ f(x) $$ approaches $$1$$. - As $$ x $$ approaches $$2$$ from the right, $$ f(x) $$ approaches $$-4$$. - At $$ x = 2 $$, $$ f(x) = 5 $$. **Points of Discontinuity:** - Vertical asymptotes at $$ x = -9 $$ and $$ x = -5 $$. - Removable discontinuity at $$ x = 2 $$. **Instructions:** 1. Draw vertical asymptotes at $$ x = -9 $$ and $$ x = -5 $$. 2. At $$ x = 2 $$, plot a point at $$ (2, 5) $$ with a hole indicating the discontinuity. 3. Sketch the graph smoothly elsewhere, ensuring the specified limits are met. --- **Summary:** The function $$ f $$ has vertical asymptotes at $$ x = -9 $$ and $$ x = -5 $$, and a removable discontinuity at $$ x = 2 $$.

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