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The given problem is to solve the limit:

\[
\lim_{{x \to \frac{1}{2}}} \frac{4x^2 - 1}{x - \frac{1}{2}}.
\]

### Step-by-Step Solution:

1. **Substitution Check:**
   First, let's directly substitute \( x = \frac{1}{2} \) into the expression:

   \[
   4\left(\frac{1}{2}\right)^2 - 1 = 4 \cdot \frac{1}{4} - 1 = 1 - 1 = 0.
   \]
   
   The denominator:
   \[
   x - \frac{1}{2} = \frac{1}{2} - \frac{1}{2} = 0.
   \]

   This gives a \(\frac{0}{0}\) indeterminate form, so we need to simplify the expression further.

2. **Factorization:**
   The numerator \( 4x^2 - 1 \) can be factored as a difference of squares:
   \[
   4x^2 - 1 = (2x - 1)(2x + 1).
   \]

   The limit now becomes:
   \[
   \lim_{{x \to \frac{1}{2}}} \frac{(2x - 1)(2x + 1)}{x - \frac{1}{2}}.
   \]

3. **Cancelation:**
   Notice that \( x - \frac{1}{2} \) is the same as \( 2x - 1 \), so we can cancel the common term from the numerator and denominator:
   \[
   \lim_{{x \to \frac{1}{2}}} \frac{(2x - 1)(2x + 1)}{2x - 1} = \lim_{{x \to \frac{1}{2}}} (2x + 1).
   \]

4. **Final Substitution:**
   Now substitute \( x = \frac{1}{2} \) into the simplified expression:
   \[
   2\left(\frac{1}{2}\right) + 1 = 1 + 1 = 2.
   \]

Thus, the value of the limit is:

\[
\boxed{2}.
\]

---

Would you like further details or have any questions?

Here are some related questions:
1. What is L'Hopital's Rule, and how can it help in solving limits with indeterminate forms?
2. How do you handle limits that result in a \(\frac{0}{0}\) form?
3. Can every indeterminate form be simplified through algebraic techniques like factoring?
4. What is the geometric interpretation of a limit?
5. How does the concept of continuity relate to limits?

**Tip:** Always check for factorizations, especially in indeterminate forms, as this often simplifies the limit evaluation.

Find the limit: lim (x -> 1/2) (4x^2 - 1) / (x - 1/2)

Learn how to solve the limit lim (x -> 1/2) (4x^2 - 1) / (x - 1/2) using factoring techniques and simplifying indeterminate forms. This step-by-step guide covers essential limit-solving methods for high school students.

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