Practice worksheet featuring two different function graphs and 20 corresponding limit problems for students to solve.

To solve the problem of finding the values of each limit using the given graphs, we need to analyze the behavior of the function \( f(x) \) as \( x \) approaches specific values from different directions. Let's go through each limit step by step.

Graph 1:

#### 1. \( \lim_{x \to -\infty} f(x) \)
- As \( x \) approaches negative infinity, the graph shows that \( f(x) \) approaches a horizontal asymptote at \( y = 0 \).
- Answer: \( \lim_{x \to -\infty} f(x) = 0 \)

#### 2. \( \lim_{x \to -3^-} f(x) \)
- As \( x \) approaches \(-3\) from the left, the graph shows that \( f(x) \) approaches \(-\infty\).
- Answer: \( \lim_{x \to -3^-} f(x) = -\infty \)

#### 3. \( \lim_{x \to -3^+} f(x) \)
- As \( x \) approaches \(-3\) from the right, the graph shows that \( f(x) \) approaches \(+\infty\).
- Answer: \( \lim_{x \to -3^+} f(x) = +\infty \)

#### 4. \( \lim_{x \to -3} f(x) \)
- Since the left-hand limit (\(-\infty\)) and the right-hand limit (\(+\infty\)) are not equal, the limit does not exist.
- Answer: \( \lim_{x \to -3} f(x) \) does not exist (DNE).

#### 5. \( \lim_{x \to 0^-} f(x) \)
- As \( x \) approaches \(0\) from the left, the graph shows that \( f(x) \) approaches \(+\infty\).
- Answer: \( \lim_{x \to 0^-} f(x) = +\infty \)

#### 6. \( \lim_{x \to 0^+} f(x) \)
- As \( x \) approaches \(0\) from the right, the graph shows that \( f(x) \) approaches \(+\infty\).
- Answer: \( \lim_{x \to 0^+} f(x) = +\infty \)

#### 7. \( \lim_{x \to 0} f(x) \)
- Since both the left-hand limit and the right-hand limit approach \(+\infty\), the limit exists and is \(+\infty\).
- Answer: \( \lim_{x \to 0} f(x) = +\infty \)

#### 8. \( \lim_{x \to 3^-} f(x) \)
- As \( x \) approaches \(3\) from the left, the graph shows that \( f(x) \) approaches \(+\infty\).
- Answer: \( \lim_{x \to 3^-} f(x) = +\infty \)

#### 9. \( \lim_{x \to 3^+} f(x) \)
- As \( x \) approaches \(3\) from the right, the graph shows that \( f(x) \) approaches \(+\infty\).
- Answer: \( \lim_{x \to 3^+} f(x) = +\infty \)

#### 10. \( \lim_{x \to 3} f(x) \)
- Since both the left-hand limit and the right-hand limit approach \(+\infty\), the limit exists and is \(+\infty\).
- Answer: \( \lim_{x \to 3} f(x) = +\infty \)

Graph 2:

#### 11. \( \lim_{x \to -\infty} f(x) \)
- As \( x \) approaches negative infinity, the graph shows that \( f(x) \) approaches a horizontal asymptote at \( y = 1 \).
- Answer: \( \lim_{x \to -\infty} f(x) = 1 \)

#### 12. \( \lim_{x \to -2^-} f(x) \)
- As \( x \) approaches \(-2\) from the left, the graph shows that \( f(x) \) approaches \(1\).
- Answer: \( \lim_{x \to -2^-} f(x) = 1 \)

#### 13. \( \lim_{x \to -2^+} f(x) \)
- As \( x \) approaches \(-2\) from the right, the graph shows that \( f(x) \) approaches \(3\).
- Answer: \( \lim_{x \to -2^+} f(x) = 3 \)

#### 14. \( \lim_{x \to -2} f(x) \)
- Since the left-hand limit (\(1\)) and the right-hand limit (\(3\)) are not equal, the limit does not exist.
- Answer: \( \lim_{x \to -2} f(x) \) does not exist (DNE).

#### 15. \( \lim_{x \to 2^-} f(x) \)
- As \( x \) approaches \(2\) from the left, the graph shows that \( f(x) \) approaches \(3\).
- Answer: \( \lim_{x \to 2^-} f(x) = 3 \)

#### 16. \( \lim_{x \to 2^+} f(x) \)
- As \( x \) approaches \(2\) from the right, the graph shows that \( f(x) \) approaches \(3\).
- Answer: \( \lim_{x \to 2^+} f(x) = 3 \)

#### 17. \( \lim_{x \to 2} f(x) \)
- Since both the left-hand limit and the right-hand limit approach \(3\), the limit exists and is \(3\).
- Answer: \( \lim_{x \to 2} f(x) = 3 \)

#### 18. \( \lim_{x \to \infty} f(x) \)
- As \( x \) approaches positive infinity, the graph shows that \( f(x) \) approaches a horizontal asymptote at \( y = 3 \).
- Answer: \( \lim_{x \to \infty} f(x) = 3 \)

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & 0 \\
2. & -\infty \\
3. & +\infty \\
4. & \text{DNE} \\
5. & +\infty \\
6. & +\infty \\
7. & +\infty \\
8. & +\infty \\
9. & +\infty \\
10. & +\infty \\
11. & 1 \\
12. & 1 \\
13. & 3 \\
14. & \text{DNE} \\
15. & 3 \\
16. & 3 \\
17. & 3 \\
18. & 3 \\
19. & 3 \\
20. & 3 \\
\end{array}
}
\]