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The image presents a mathematical limit problem. Let's extract and solve it:

\[
\lim_{n \to 1} \left( \frac{1}{1 - x} + \frac{1}{1 - x^3} \right)
\]

To solve this limit as \( x \to 1 \):

1. **Direct Substitution**:
   Substituting \( x = 1 \) into the expression:
   
   \[
   \frac{1}{1 - 1} + \frac{1}{1 - 1^3} = \frac{1}{0} + \frac{1}{0}
   \]

   This results in an indeterminate form \( \frac{1}{0} \), which suggests the function has a singularity (division by zero) at \( x = 1 \). Therefore, we must analyze the behavior of the function around \( x = 1 \) to understand its limits properly.

2. **Factorization and Simplification**:
   We need to approach \( x \) from values near but not equal to 1. Let's analyze the limit behavior by simplifying and factoring.

For small deviations around \( x = 1 \), the terms behave in such a way that the function approaches infinity as \( x \) nears 1 from either side. The individual terms \(\frac{1}{1 - x}\) and \(\frac{1}{1 - x^3}\) both exhibit singular behavior near \( x = 1 \), driving the overall expression to positive or negative infinity.

Thus, the limit does **not exist** in the traditional sense as \( x \to 1 \) because the function blows up.

Let me know if you'd like further clarifications!

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Would you like to explore related concepts or examples? Here are 5 questions that build on this idea:

1. How do we handle indeterminate forms like \( \frac{1}{0} \) in limit problems?
2. What are common techniques for solving limits with singularities?
3. How can L'Hopital's Rule help when facing indeterminate limits?
4. How can limits be used to define continuity at a point?
5. What are the implications of a limit tending to infinity in real-world applications?

**Tip**: For limit problems involving singularities or undefined points, always check if simplification, factorization, or L'Hopital's Rule can be applied to resolve the issue.

lim_{x → 1} (1 / (1 - x) + 1 / (1 - x^3))

Learn how to solve the limit problem lim_{x → 1} (1/(1 - x) + 1/(1 - x^3)). This problem explores concepts like singularities and indeterminate forms, with a focus on analyzing the behavior of functions near undefined points. Suitable for advanced high school students.

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