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Let's analyze the calculus quiz step by step, focusing first on the limit problems based on the provided graph of the function \( f(x) \).

### Limits from the graph:
1. **\(\lim_{x \to 1} f(x)\)**:  
   From the graph, as \(x\) approaches 1 from both sides, the function appears to approach a value around 2.  
   \[ \lim_{x \to 1} f(x) = 2 \]

2. **\(\lim_{x \to 2} f(x)\)**:  
   At \(x = 2\), there is a vertical asymptote, and the function seems to approach \(-\infty\) from both sides.  
   \[ \lim_{x \to 2} f(x) = -\infty \]

3. **\(\lim_{x \to -\infty} f(x)\)**:  
   As \(x\) approaches \(-\infty\), the function seems to approach 0 from above.  
   \[ \lim_{x \to -\infty} f(x) = 0 \]

4. **\(\lim_{x \to \infty} f(x)\)**:  
   As \(x\) approaches \(+\infty\), the function appears to stabilize at a value of 1.  
   \[ \lim_{x \to \infty} f(x) = 1 \]

5. **\(\lim_{x \to 3^+} f(x)\)**:  
   As \(x\) approaches 3 from the right side, the function seems to approach 4.  
   \[ \lim_{x \to 3^+} f(x) = 4 \]

6. **\(\lim_{x \to 3^-} f(x)\)**:  
   As \(x\) approaches 3 from the left side, the function seems to approach 0.  
   \[ \lim_{x \to 3^-} f(x) = 0 \]

### Limits without L'Hopital's Rule:

7. **\(\lim_{x \to \infty} \frac{7x^{5/2}}{8x^3 - 4x - 2}\)**:  
   As \(x \to \infty\), we compare the degrees of the polynomial. The degree of the numerator is \( \frac{5}{2} \) and the degree of the denominator is 3. Since the degree of the denominator is higher, the limit will be 0.  
   \[ \lim_{x \to \infty} \frac{7x^{5/2}}{8x^3 - 4x - 2} = 0 \]

8. **\(\lim_{x \to 3^+} \frac{-6}{2x - 6}\)**:  
   For \(x = 3\), the denominator becomes 0, so the limit would involve division by zero. However, for values slightly larger than 3, the denominator is positive. Therefore, the expression tends to \(-\infty\) as \(x \to 3^+\).  
   \[ \lim_{x \to 3^+} \frac{-6}{2x - 6} = -\infty \]

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Would you like more details on any of these solutions or the methodology used?

### Related Questions:
1. What happens when the degrees of the numerator and denominator in a rational function are equal?
2. How do vertical asymptotes affect the limits of functions?
3. What is the difference between limits approaching from the left (\(x \to c^-\)) versus the right (\(x \to c^+\))?
4. How can one identify horizontal asymptotes from a function's graph?
5. What is the behavior of polynomials as \(x \to \infty\) or \(x \to -\infty\)?

### Tip:
When finding limits from a graph, always examine the function's behavior as \(x\) approaches the point of interest from both directions to determine if the limit exists.

Use the graph of the function f(x) to determine each limit or value. If the limit does not exist, write 'dne'. Also, find the specified limits without using L'Hopital's Rule.

Solve calculus limit problems using the graph of a function. This problem covers key concepts like vertical asymptotes, horizontal asymptotes, and the limit laws. Also, find limits of rational functions without using L'Hopital's Rule, ideal for undergraduate calculus students.

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