https://file.aiquickdraw.com/mathbot/file/1726300410849ezsy71np.jpg

Let's analyze the two limits presented in the image.

### 1. Limit 1:

\[
\lim_{x \to -1} \frac{x^{15} + 1}{x^9 + 1}
\]

To solve this, let's substitute \( x = -1 \) directly:

\[
\frac{(-1)^{15} + 1}{(-1)^9 + 1} = \frac{-1 + 1}{-1 + 1} = \frac{0}{0}
\]

Since this results in the indeterminate form \( \frac{0}{0} \), we can apply **L'Hopital's Rule**, which involves differentiating the numerator and denominator.

- The derivative of the numerator \( x^{15} + 1 \) is \( 15x^{14} \).
- The derivative of the denominator \( x^9 + 1 \) is \( 9x^8 \).

Now, applying L'Hopital's Rule:

\[
\lim_{x \to -1} \frac{15x^{14}}{9x^8} = \frac{15(-1)^{14}}{9(-1)^8} = \frac{15}{9} = \frac{5}{3}
\]

Thus, the value of the limit is:

\[
\boxed{\frac{5}{3}}
\]

### 2. Limit 2:

\[
\lim_{x \to a} \frac{x^{\frac{2}{7}} - a^{\frac{2}{7}}}{x - a}
\]

This limit is in a form suitable for the **derivative definition**, where:

\[
\lim_{x \to a} \frac{f(x) - f(a)}{x - a}
\]

The function \( f(x) = x^{\frac{2}{7}} \). The limit can be evaluated as the derivative of \( f(x) \) at \( x = a \).

The derivative of \( f(x) = x^{\frac{2}{7}} \) is:

\[
f'(x) = \frac{2}{7}x^{-\frac{5}{7}}
\]

Evaluating this derivative at \( x = a \):

\[
f'(a) = \frac{2}{7}a^{-\frac{5}{7}}
\]

Thus, the value of the second limit is:

\[
\boxed{\frac{2}{7} a^{-\frac{5}{7}}}
\]

---

Would you like more details on either limit or the steps involved?

Here are 5 related questions for further exploration:
1. How is L'Hopital's Rule derived and when can it be applied?
2. What is the geometric interpretation of a limit?
3. How can we apply Taylor series to approximate limits near indeterminate forms?
4. Why is the derivative definition useful in evaluating limits?
5. What happens if a limit does not exist, and how can we identify such cases?

**Tip:** Always check for indeterminate forms like \( \frac{0}{0} \) or \( \infty/\infty \) before applying L'Hopital's Rule!

What are the limits of the following functions:
1) lim x → -1 (x^15 + 1) / (x^9 + 1)
2) lim x → a (x^(2/7) - a^(2/7)) / (x - a)?

This page provides step-by-step solutions for evaluating two limits: one using L'Hopital's Rule and the other using the definition of the derivative. Learn how to handle indeterminate forms like 0/0 and apply calculus techniques.

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