https://file.aiquickdraw.com/mathbot/file/17254628982067qtnu1wo.jpg

Let's evaluate the limits provided in the image step by step.

### Problem a) \(\lim_{x \to 1} (2x^2 - 1)\)

1. **Substitute \(x = 1\) into the function:**
   \[
   2(1)^2 - 1 = 2(1) - 1 = 2 - 1 = 1
   \]

   **Answer:** The limit is \(\boxed{1}\).

### Problem b) \(\lim_{x \to 1} (2x + 1)(3 - 2x)\)

1. **Substitute \(x = 1\) into the function:**
   \[
   (2(1) + 1)(3 - 2(1)) = (2 + 1)(3 - 2) = 3 \times 1 = 3
   \]

   **Answer:** The limit is \(\boxed{3}\).

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Would you like more details on how to solve these types of problems or have any questions?

Here are some related questions you might find interesting:
1. How do you handle limits that result in indeterminate forms like \(\frac{0}{0}\)?
2. What are the common limit laws and how can they simplify limit calculations?
3. Can you explain the Squeeze Theorem and give an example where it's useful?
4. How do you evaluate limits at infinity, especially for rational functions?
5. What is L'Hôpital's rule, and when can it be applied to find limits?

**Tip:** Always check if direct substitution works for limits before considering other techniques like factoring or L'Hôpital's rule.

Evaluate the following limits: a) lim_{x -> 1} (2x^2 - 1), b) lim_{x -> 1} (2x + 1)(3 - 2x)

Learn how to evaluate limits using direct substitution with the examples lim_{x -> 1} (2x^2 - 1) and lim_{x -> 1} (2x + 1)(3 - 2x). This problem covers key concepts in algebra and limits, suitable for Grades 11-12 students.

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