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We are tasked with finding the limit:

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

### Step 1: Factor the denominator
Notice that the denominator can be factored as a difference of squares:

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

So, the expression becomes:

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

### Step 2: Direct substitution
If we substitute \(x = 1\) directly into the expression, we get:

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

Since the denominator becomes zero, we are dealing with an indeterminate form (specifically division by zero), and we must check the behavior of the function as \(x\) approaches 1 from both sides.

### Step 3: Analyze the limit from both sides
- **As \(x \to 1^+\)** (from the right):
  The term \(x - 1\) is positive and small, while \(x + 1\) is positive. Hence, the overall fraction tends to \(+\infty\).

- **As \(x \to 1^-\)** (from the left):
  The term \(x - 1\) is negative and small, while \(x + 1\) is positive. Hence, the overall fraction tends to \(-\infty\).

Since the limits from the left and right are not equal, the limit **does not exist**.

### Final Answer:
The correct option is **b. No existe**.

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Would you like more details or have any questions about this?

Here are five related questions to expand your understanding:
1. What is the difference between one-sided and two-sided limits?
2. How do you factor more complex quadratic expressions?
3. What are the most common forms of indeterminate limits, and how are they resolved?
4. How can L'Hopital's Rule be applied to limits involving indeterminate forms?
5. How do we evaluate limits approaching infinity or negative infinity?

**Tip**: Always attempt to factor or simplify expressions when facing limits involving indeterminate forms to avoid errors in direct substitution.

Explore how to find the limit of the expression x / (x^2 - 1) as x approaches 1. This problem involves factorization and analyzing limits approaching an indeterminate form. Perfect for students in advanced math courses, Grades 11-12.

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