https://r2.erweima.ai/mathbot/image/1725158182636jsvl6uqt.jpg

The image contains three limit problems to solve. Let's go through each one:

1. **Problem 6**:  
   \[
   \lim_{{x \to 1}} (x^2 - 1)
   \]
   - **Solution**:  
     To solve this limit, substitute \( x = 1 \) directly into the expression:
     \[
     \lim_{{x \to 1}} (x^2 - 1) = 1^2 - 1 = 0
     \]

2. **Problem 7**:  
   \[
   \lim_{{x \to -1}} \frac{{(x^2 + 1)}}{x}
   \]
   - **Solution**:  
     Again, substitute \( x = -1 \) directly:
     \[
     \lim_{{x \to -1}} \frac{{x^2 + 1}}{x} = \frac{{(-1)^2 + 1}}{-1} = \frac{1 + 1}{-1} = \frac{2}{-1} = -2
     \]

3. **Problem 8**:  
   \[
   \lim_{{x \to 3}} \frac{x^2 - 9}{x - 3}
   \]
   - **Solution**:  
     This expression results in an indeterminate form \( \frac{0}{0} \) if you directly substitute \( x = 3 \). So, we need to factor the numerator:
     \[
     \frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3}
     \]
     Cancel the common factor \( (x - 3) \):
     \[
     \lim_{{x \to 3}} (x + 3) = 3 + 3 = 6
     \]

If you have any questions or need more details on any step, feel free to ask!

---

Here are five related questions to explore further:

1. What are the basic rules for evaluating limits?
2. How can you recognize an indeterminate form when calculating a limit?
3. How does L'Hôpital's rule apply to limits?
4. What are the conditions for using direct substitution in limit problems?
5. How do you handle limits that approach infinity?

**Tip**: When solving limits that result in indeterminate forms, factoring or using algebraic manipulation can often simplify the expression and resolve the limit.

Explore detailed solutions to limit problems including direct substitution and algebraic manipulation. Learn about recognizing indeterminate forms and resolving limits effectively. Suitable for advanced high school students and anyone preparing for calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation