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Let's evaluate the given limit expression step by step. The expression is:

\[
\lim_{{x \to 0}} \frac{x}{{\frac{1}{{2 + x}} - \frac{1}{2}}}
\]

### Step 1: Simplify the denominator
The denominator is a difference of two fractions, \(\frac{1}{{2 + x}} - \frac{1}{2}\). To subtract these fractions, we need to find a common denominator:

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

So, the expression becomes:

\[
\lim_{{x \to 0}} \frac{x}{\frac{-x}{2(2 + x)}}
\]

### Step 2: Simplify the fraction
Now, simplify the complex fraction by multiplying \(x\) by the reciprocal of \(\frac{-x}{2(2 + x)}\):

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

### Step 3: Evaluate the limit
Now, take the limit as \(x\) approaches 0:

\[
\lim_{{x \to 0}} -2(2 + x) = -2(2 + 0) = -4
\]

Thus, the correct value of the limit is:

\[
\boxed{-4}
\]

The correct answer is \(-4\), not \(\frac{1}{4}\).

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Would you like any further explanation or more details?

Here are 5 related questions for further practice:
1. What is the limit of \(\lim_{{x \to 0}} \frac{x}{{x + 1}}\)?
2. How would you evaluate \(\lim_{{x \to 0}} \frac{x}{{(x^2 + 1) - 1}}\)?
3. Can you solve \(\lim_{{x \to 0}} \frac{\sin(x)}{x}\)?
4. How do you handle indeterminate forms when evaluating limits?
5. What is the limit of \(\lim_{{x \to 1}} \frac{1 - \cos(x)}{x - 1}\)?

**Tip**: Always simplify the expression as much as possible before applying the limit, and check for indeterminate forms like \(\frac{0}{0}\) where you might need additional techniques such as factoring or L'Hopital's Rule.

Evaluate the limit: lim_{x -> 0} x / ( (1/(2 + x)) - (1/2) )

Learn how to evaluate the limit lim_{x -> 0} x / ( (1/(2 + x)) - (1/2) ). This problem demonstrates important concepts of algebraic manipulation and limits, focusing on handling fractions and evaluating indeterminate forms. Suitable for college-level calculus students.

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