https://file.aiquickdraw.com/mathbot/file/1725920588017mp301bk7.jpg

Let's analyze the given quiz based on the image, focusing on determining the limits from the provided graph and solving the limit problems.

### First, we'll tackle the graph-related limits:
#### Using the graph of \( f(x) \), let's determine the following limits:

1. \( \lim_{x \to 1} f(x) \):
   - At \( x = 1 \), observe that the graph shows a jump discontinuity. The left-hand limit (\( x \to 1^{-} \)) seems to approach \( 2 \), while the right-hand limit (\( x \to 1^{+} \)) appears to approach \( 4 \).
   - Since the left-hand and right-hand limits are different, **the limit does not exist** (DNE).

2. \( \lim_{x \to 2} f(x) \):
   - At \( x = 2 \), the graph smoothly approaches a value from both sides. Both the left-hand and right-hand limits approach \( 2 \).
   - So, the limit is \( \boxed{2} \).

3. \( \lim_{x \to -\infty} f(x) \):
   - As \( x \) tends to negative infinity, the function \( f(x) \) appears to approach \( -3 \) (based on the horizontal asymptote at \( y = -3 \)).
   - So, the limit is \( \boxed{-3} \).

4. \( \lim_{x \to \infty} f(x) \):
   - As \( x \) tends to positive infinity, the function \( f(x) \) appears to approach \( 0 \) (horizontal asymptote at \( y = 0 \)).
   - So, the limit is \( \boxed{0} \).

5. \( \lim_{x \to 1^{-}} f(x) \):
   - The left-hand limit as \( x \) approaches \( 1 \) from the left (\( x \to 1^{-} \)) is clearly \( 2 \).
   - So, the limit is \( \boxed{2} \).

6. \( \lim_{x \to 1^{+}} f(x) \):
   - The right-hand limit as \( x \) approaches \( 1 \) from the right (\( x \to 1^{+} \)) is clearly \( 4 \).
   - So, the limit is \( \boxed{4} \).

### Now, let's solve the algebraic limits:

7. \( \lim_{x \to \infty} \frac{7x^{5/2}}{8x^3 - 4x - 2} \)
   - When \( x \to \infty \), the term with the highest power of \( x \) dominates in both the numerator and denominator.
   - In the numerator, the term is \( 7x^{5/2} \), and in the denominator, the term is \( 8x^3 \).
   - Simplifying: \( \frac{7x^{5/2}}{8x^3} = \frac{7}{8x^{1/2}} \).
   - As \( x \to \infty \), \( x^{1/2} \to \infty \), so \( \frac{7}{8x^{1/2}} \to 0 \).
   - Therefore, the limit is \( \boxed{0} \).

8. \( \lim_{x \to 3^{+}} \frac{-6}{2x - 6} \)
   - As \( x \to 3^{+} \), the denominator \( 2x - 6 \) becomes \( 0^{+} \), meaning it approaches 0 from the positive side.
   - So, \( \frac{-6}{0^{+}} \) tends to \( -\infty \).
   - Therefore, the limit is \( \boxed{-\infty} \).

### Summary of Answers:
1. \( \lim_{x \to 1} f(x) \): DNE  
2. \( \lim_{x \to 2} f(x) \): 2  
3. \( \lim_{x \to -\infty} f(x) \): -3  
4. \( \lim_{x \to \infty} f(x) \): 0  
5. \( \lim_{x \to 1^{-}} f(x) \): 2  
6. \( \lim_{x \to 1^{+}} f(x) \): 4  
7. \( \lim_{x \to \infty} \frac{7x^{5/2}}{8x^3 - 4x - 2} \): 0  
8. \( \lim_{x \to 3^{+}} \frac{-6}{2x - 6} \): -∞  

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### Related Questions:
1. What is the significance of horizontal asymptotes in limit problems?
2. How do we evaluate one-sided limits from graphs?
3. Why does the highest power of \( x \) dominate in limits at infinity?
4. How are limits affected by vertical asymptotes?
5. What does it mean when a limit approaches \( -\infty \)?

### Tip:
When dealing with limits at infinity, focus on the highest powers of \( x \) in both the numerator and denominator for large \( x \).

Solve the given calculus problems related to limits, both graphically and algebraically.

Explore step-by-step solutions to a Calculus I quiz covering limits from graphs and algebraic expressions. Learn how to determine limits graphically, handle discontinuities, and solve algebraic limits without using L'Hôpital's rule. Suitable for college-level calculus students.

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