https://file.aiquickdraw.com/mathbot/file/17267461416099d3765l3.jpg

I can help solve the problems shown in the image. Let's break them down one by one:

1. **Limit:**
   \[
   \lim_{x \to -1} \frac{x^2 - x}{x^3 - 1}
   \]

2. **Trigonometric expression:**
   \[
   \tan[\sec^{-1}(\sqrt{2})]
   \]

3. **Integral:**
   \[
   \int \sin(x) \, dx
   \]

### 1. Limit:
\[
\lim_{x \to -1} \frac{x^2 - x}{x^3 - 1}
\]
First, substitute \( x = -1 \) into the function:

\[
\frac{(-1)^2 - (-1)}{(-1)^3 - 1} = \frac{1 + 1}{-1 - 1} = \frac{2}{-2} = -1
\]

So, the value of the limit is:
\[
\boxed{-1}
\]

### 2. Trigonometric Expression:
\[
\tan[\sec^{-1}(\sqrt{2})]
\]
We know that \( \sec^{-1}(y) \) is the angle \( \theta \) such that \( \sec(\theta) = y \). So for \( \sec^{-1}(\sqrt{2}) \), \( \sec(\theta) = \sqrt{2} \).

From the identity:
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]
If \( \sec(\theta) = \sqrt{2} \), then \( \cos(\theta) = \frac{1}{\sqrt{2}} \), which corresponds to an angle \( \theta = \frac{\pi}{4} \).

Now, calculate \( \tan(\theta) \):
\[
\tan\left(\frac{\pi}{4}\right) = 1
\]

Thus, the result is:
\[
\boxed{1}
\]

### 3. Integral:
\[
\int \sin(x) \, dx
\]
The integral of \( \sin(x) \) is:
\[
-\cos(x) + C
\]

Thus, the result of the integral is:
\[
\boxed{-\cos(x) + C}
\]

---

Let me know if you would like further details on any part. Here are 5 related questions you can explore:

1. How do you handle limits when the function involves indeterminate forms like \( \frac{0}{0} \)?
2. What is the geometric meaning of \( \tan[\sec^{-1}(x)] \)?
3. Can you derive the integral of other basic trigonometric functions like \( \cos(x) \) or \( \tan(x) \)?
4. How do you use L'Hopital's rule to solve more complex limits?
5. What is the significance of the constant \( C \) in indefinite integrals?

**Tip:** Always check for common trigonometric identities and limit laws when solving problems, as they can simplify your calculations significantly!

Solve the following:
1. lim x → -1 (x² - x) / (x³ - 1)
2. tan(sec⁻¹(√2))
3. ∫ sin(x) dx

Learn how to solve limit problems, evaluate inverse trigonometric expressions, and compute integrals like ∫ sin(x) dx. Detailed explanations covering limits as x approaches -1, secant inverse, and basic integral techniques. Suitable for Grades 10-12 students.

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