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We are given the limit expression:

\[
\lim_{x \to \infty} x \left( \sqrt{4 + \frac{1}{x}} - 2 \right)
\]

### Step-by-Step Solution:

1. **Simplify the expression inside the square root**:
   As \( x \to \infty \), the term \( \frac{1}{x} \) approaches 0. So, we can analyze the behavior of the square root term:
   \[
   \sqrt{4 + \frac{1}{x}} \to \sqrt{4} = 2
   \]
   However, we need to calculate the behavior of the expression more carefully for large values of \( x \), so we'll expand it using a binomial approximation.

2. **Use binomial approximation**:
   For large \( x \), we can use the binomial approximation for square roots:
   \[
   \sqrt{1 + z} \approx 1 + \frac{z}{2} \quad \text{for small } z
   \]
   Here, \( z = \frac{1}{4x} \), so:
   \[
   \sqrt{4 + \frac{1}{x}} = 2 \sqrt{1 + \frac{1}{4x}} \approx 2 \left(1 + \frac{1}{8x}\right) = 2 + \frac{1}{4x}
   \]

3. **Substitute into the original expression**:
   Now substitute this approximation into the limit expression:
   \[
   x \left( \sqrt{4 + \frac{1}{x}} - 2 \right) \approx x \left( 2 + \frac{1}{4x} - 2 \right) = x \left( \frac{1}{4x} \right)
   \]
   Simplifying further:
   \[
   x \times \frac{1}{4x} = \frac{1}{4}
   \]

4. **Take the limit**:
   Since the expression simplifies to a constant value \( \frac{1}{4} \), the limit is:
   \[
   \lim_{x \to \infty} x \left( \sqrt{4 + \frac{1}{x}} - 2 \right) = \frac{1}{4}
   \]

### Final Answer:
\[
\boxed{\frac{1}{4}}
\]

Would you like further details on any step, or have more questions? Here are some related problems you could explore:

1. What if the limit was \( \lim_{x \to 0} x \left( \sqrt{4 + \frac{1}{x}} - 2 \right) \)?
2. How does the binomial approximation apply in different contexts for small values of \( z \)?
3. What would happen if the term inside the square root was more complex, like \( \sqrt{4 + \frac{1}{x^2}} \)?
4. Can you find an alternative method for solving this limit, such as L'Hopital's Rule?
5. How can we use Taylor series to approximate square roots more generally?

**Tip**: Binomial expansions are very useful for approximating functions for large or small values of variables. Familiarize yourself with these expansions for common functions!

Explore how to solve the limit lim(x → ∞) x(sqrt(4 + 1/x) - 2) using binomial approximation. This calculus problem involves understanding asymptotic behavior and applying the binomial theorem. Suitable for college-level calculus students.

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