https://file.aiquickdraw.com/mathbot/file/1724497175285wl7i1y5f.jpg

The image you provided contains some mathematical limits that need to be evaluated. I will analyze and solve the limits based on what is visible in the image.

### Analysis
The image seems to include several limit problems. Here's the structure:

1. The first limit involves a rational expression:
   \[
   \lim_{x \to \infty} \frac{x^n - ax^{n-1} - \dots - a_0}{bx^m - b_1x^{m-1} - \dots - b_0}
   \]
   It appears that the limit is evaluated for \( x \) approaching infinity.

2. The second limit also involves a rational expression:
   \[
   \lim_{x \to 0} \frac{x^3 - x}{x - 2} 
   \]

3. The third limit shown appears to involve a trigonometric function:
   \[
   \lim_{x \to 0} \frac{\sin(x)}{x}
   \]

### Solutions

#### 1. \(\lim_{x \to \infty} \frac{x^n - ax^{n-1} - \dots - a_0}{bx^m - b_1x^{m-1} - \dots - b_0}\)

- **Case 1:** If \(n > m\), the limit is \(\infty\).
- **Case 2:** If \(n < m\), the limit is \(0\).
- **Case 3:** If \(n = m\), the limit is \(\frac{1}{b}\).

#### 2. \(\lim_{x \to 0} \frac{x^3 - x}{x - 2}\)

For small \( x \), the expression simplifies as follows:

\[
\lim_{x \to 0} \frac{x(x^2 - 1)}{x - 2} = \frac{0}{-2} = 0
\]

So, the limit is 0.

#### 3. \(\lim_{x \to 0} \frac{\sin(x)}{x}\)

This is a well-known limit in calculus:

\[
\lim_{x \to 0} \frac{\sin(x)}{x} = 1
\]

### Summary of Answers:

1. The first limit depends on the relationship between \(n\) and \(m\).
2. The second limit as \(x \to 0\) evaluates to \(0\).
3. The third limit evaluates to \(1\).

Would you like further details, or do you have any questions?

Here are five related questions:
1. How do you evaluate limits of rational functions as \( x \) approaches infinity?
2. What are common indeterminate forms, and how are they resolved?
3. What are the fundamental trigonometric limits?
4. How can L'Hôpital's Rule be applied to evaluate limits?
5. How do you handle limits involving piecewise functions?

**Tip:** For rational functions, always compare the highest powers in the numerator and denominator to simplify the limit evaluation.

Learn how to evaluate limits of rational expressions and trigonometric functions with detailed examples and solutions. This page covers key concepts in calculus, including limit properties and L'Hôpital's Rule. Suitable for advanced high school students and above.

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