https://r2.erweima.ai/mathbot/image/1725764356683a6w2h9ut.jpg

The expression shown in the image is the following limit problem:

\[
\lim_{x \to 0} \frac{2 - \sqrt{4 - x}}{x}
\]

This limit can be solved by rationalizing the numerator. Here's how to proceed:

### Step 1: Multiply by the conjugate
To simplify the expression, we multiply both the numerator and denominator by the conjugate of the numerator, \( 2 + \sqrt{4 - x} \). This will help eliminate the square root:

\[
\frac{2 - \sqrt{4 - x}}{x} \times \frac{2 + \sqrt{4 - x}}{2 + \sqrt{4 - x}} = \frac{(2 - \sqrt{4 - x})(2 + \sqrt{4 - x})}{x(2 + \sqrt{4 - x})}
\]

### Step 2: Simplify the numerator
Using the difference of squares formula:

\[
(2 - \sqrt{4 - x})(2 + \sqrt{4 - x}) = 2^2 - (\sqrt{4 - x})^2 = 4 - (4 - x) = x
\]

Thus, the expression simplifies to:

\[
\frac{x}{x(2 + \sqrt{4 - x})}
\]

### Step 3: Cancel \( x \) in the numerator and denominator
Since \( x \neq 0 \), we can cancel \( x \) in both the numerator and denominator:

\[
\frac{1}{2 + \sqrt{4 - x}}
\]

### Step 4: Evaluate the limit as \( x \to 0 \)
Now substitute \( x = 0 \) into the simplified expression:

\[
\frac{1}{2 + \sqrt{4 - 0}} = \frac{1}{2 + 2} = \frac{1}{4}
\]

Thus, the value of the limit is:

\[
\lim_{x \to 0} \frac{2 - \sqrt{4 - x}}{x} = \frac{1}{4}
\]

### Conclusion:
The limit evaluates to \( \frac{1}{4} \).

---

Would you like more details on any part of the solution? Here are some related questions to explore:

1. Why is multiplying by the conjugate helpful in limits involving square roots?
2. Can you apply the L'Hopital's rule to solve this limit as well?
3. What happens if you evaluate the limit directly without simplification?
4. How can you recognize when to rationalize the numerator in similar problems?
5. How does the difference of squares formula work in simplifying expressions?

**Tip:** Rationalizing the numerator is a common technique for solving limits with square roots.

Evaluate the limit: \( \lim_{x \to 0} \frac{2 - \sqrt{4 - x}}{x} \)

Learn how to evaluate the limit \( \lim_{x \to 0} \frac{2 - \sqrt{4 - x}}{x} \) using rationalization. This problem involves algebraic techniques such as the difference of squares and is suitable for students learning calculus.

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