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The given expression represents the following limit:

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

### Steps to solve:

1. **Direct substitution**:
   Substituting \(x = 1\) directly into the function:
   \[
   \frac{3(1)^2 - 1}{\sqrt{1} - 1} = \frac{3 - 1}{1 - 1} = \frac{2}{0}
   \]
   This gives an undefined form, which suggests the need for algebraic manipulation.

2. **Rationalize the denominator**:
   To resolve this, multiply both the numerator and denominator by the conjugate of the denominator \( \sqrt{x} + 1 \):
   \[
   \frac{3x^2 - 1}{\sqrt{x} - 1} \times \frac{\sqrt{x} + 1}{\sqrt{x} + 1}
   \]
   This gives:
   \[
   \frac{(3x^2 - 1)(\sqrt{x} + 1)}{(\sqrt{x} - 1)(\sqrt{x} + 1)} = \frac{(3x^2 - 1)(\sqrt{x} + 1)}{x - 1}
   \]

3. **Factoring the numerator**:
   Factor \(3x^2 - 1\) as:
   \[
   3x^2 - 1 = (x - 1)(3x + 1)
   \]
   Now, the expression becomes:
   \[
   \frac{(x - 1)(3x + 1)(\sqrt{x} + 1)}{x - 1}
   \]
   Cancel the common factor \(x - 1\):
   \[
   (3x + 1)(\sqrt{x} + 1)
   \]

4. **Substitute \(x = 1\)**:
   After simplifying, substitute \(x = 1\) into the expression:
   \[
   (3(1) + 1)(\sqrt{1} + 1) = (3 + 1)(1 + 1) = 4 \times 2 = 8
   \]

Thus, the limit is:
\[
\boxed{8}
\]

Would you like any further clarification or details on any of the steps? Here are some related questions for further practice:

1. What is the significance of rationalizing the denominator in limit problems?
2. How do you handle limits that result in an indeterminate form?
3. Can the factorization technique be applied in other types of limits?
4. What are some alternative strategies if rationalizing doesn't simplify the limit?
5. How does L'Hopital's Rule apply to limits with \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) forms?

**Tip:** Always check for factorization opportunities when direct substitution leads to an indeterminate form like \( \frac{0}{0} \).

Find the limit of the expression lim(x -> 1) (3x^2 - 1) / (√x - 1).

Learn how to evaluate the limit of the expression (3x^2 - 1) / (√x - 1) as x approaches 1. This problem covers limit concepts, rationalization techniques, and factoring methods, suitable for high school or college-level calculus students.

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