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Let's solve the given limit:

\[
\lim_{x \to 1} \frac{x^3 + 1}{x^2 - 1}
\]

### Step 1: Substitute \( x = 1 \)

First, check if directly substituting \( x = 1 \) leads to an indeterminate form:

\[
\frac{1^3 + 1}{1^2 - 1} = \frac{1 + 1}{1 - 1} = \frac{2}{0}
\]

This leads to a division by zero, so we need to simplify the expression.

### Step 2: Factor the numerator and denominator

- The numerator \( x^3 + 1 \) is a sum of cubes, which can be factored as:
  
  \[
  x^3 + 1 = (x + 1)(x^2 - x + 1)
  \]

- The denominator \( x^2 - 1 \) is a difference of squares, which can be factored as:

  \[
  x^2 - 1 = (x - 1)(x + 1)
  \]

### Step 3: Simplify the expression

Substituting the factored forms into the limit:

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

We can cancel the common factor \( (x + 1) \) from both the numerator and denominator:

\[
\lim_{x \to 1} \frac{x^2 - x + 1}{x - 1}
\]

### Step 4: Apply L'Hôpital's Rule (Optional)

At this point, directly substituting \( x = 1 \) into the simplified expression still results in \( \frac{1 - 1 + 1}{1 - 1} = \frac{1}{0} \), which is undefined. However, we can try applying **L'Hôpital's Rule** because the original limit gives a \( \frac{0}{0} \) indeterminate form.

#### Differentiate the numerator and denominator:

- The derivative of the numerator \( x^2 - x + 1 \) is \( 2x - 1 \).
- The derivative of the denominator \( x - 1 \) is 1.

Now, we compute the limit of the derivatives:

\[
\lim_{x \to 1} \frac{2x - 1}{1}
\]

Substituting \( x = 1 \) into the new expression:

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

### Final Answer:

The limit is \( \boxed{1} \).

Would you like more details or have any questions?

### Related Questions:
1. What is L'Hôpital's Rule and when can it be applied?
2. How do you factor the sum of cubes like \( x^3 + 1 \)?
3. What are the common indeterminate forms in calculus?
4. Can every limit that results in \( \frac{0}{0} \) be solved by factoring?
5. How do you determine if direct substitution in limits is possible?

### Tip:
Always check for indeterminate forms before simplifying or applying L'Hôpital's Rule in limits.

El resultado del siguiente límite es: lim (x -> 1) (x^3 + 1) / (x^2 - 1)

Learn how to solve the limit lim (x -> 1) (x^3 + 1) / (x^2 - 1) using factoring techniques and L'Hôpital's Rule. This step-by-step explanation covers key calculus concepts such as sum of cubes, difference of squares, and handling indeterminate forms.

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