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The image contains a portion of a calculus exam. From what I can see, the tasks involve evaluating limits and finding derivatives of given functions. I'll extract the details from the image for clarity:

### Problems:
1. **Limits**:
   \[
   \lim_{{x \to 1}} \frac{{x^3 - 1}}{{x^2 - 1}}, \quad \lim_{{x \to 1}} \frac{{4x^2 + 2}}{{3x^3 - 5}}, \quad \lim_{{x \to \infty}} \frac{{3 - x}}{{9x^2 + 5x}}
   \]

2. **Derivatives**:
   Evaluate the derivatives of the following functions:
   a. \( f(x) = \sqrt[3]{{x}} (x^3 + 4) + \frac{{3x}}{{3x^2 + 1}} \)  
   b. \( f(x) = \frac{{x^3 - 3x^2}}{{(2x - 3)^2}} \)  
   c. \( f(x) = \left(\frac{{3x}}{{x^2 + 1}}\right)^5 \)

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Would you like detailed solutions for each of the limits and derivatives?

Here are five related questions for further practice:
1. How do you solve indeterminate forms using L'Hopital's Rule?
2. What is the process for finding derivatives using the quotient rule?
3. How do you apply the chain rule when differentiating a composition of functions?
4. Can you explain how limits at infinity affect the behavior of rational functions?
5. How do you find the derivative of a product involving a power function and a polynomial?

**Tip:** Always simplify the function before applying differentiation rules, as it often makes the process easier and reduces chances of errors.

Evaluate the given limits and derivatives as shown on the calculus exam. The tasks include calculating limits and derivatives for various functions, involving limits at infinity and applying derivative rules.

Learn how to evaluate limits and derivatives, including limits at infinity and indeterminate forms, using L'Hopital's Rule and the Quotient Rule. This calculus problem is ideal for university-level students taking Calculus I.

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