Question

The solution to the limit \( \lim_{x \to 0} \frac{2 - \sqrt{4 - x}}{x} \) is ...........

Hint:

Rationalizing the numerator is a helpful technique when dealing with square roots in limits.

Answer 1

Correct Answer: 0.25

Step 1: Rewriting the limit expression.
We are tasked with evaluating the limit: \[ \lim_{x \to 0} \frac{2 - \sqrt{4 - x}}{x}. \]

Step 2: Rationalizing the numerator.
To simplify the expression, we multiply the numerator and the denominator by the conjugate of the numerator: \[ \frac{2 - \sqrt{4 - x}}{x} \times \frac{2 + \sqrt{4 - x}}{2 + \sqrt{4 - x}} = \frac{(2 - \sqrt{4 - x})(2 + \sqrt{4 - x})}{x(2 + \sqrt{4 - x})}. \]

Step 3: Simplifying the expression.
Using the identity \( (a - b)(a + b) = a^2 - b^2 \), we get: \[ \frac{4 - (4 - x)}{x(2 + \sqrt{4 - x})} = \frac{x}{x(2 + \sqrt{4 - x})}. \] Simplifying further: \[ \frac{1}{2 + \sqrt{4 - x}}. \]

Step 4: Evaluating the limit as \( x \to 0 \).
As \( x \to 0 \), \( \sqrt{4 - x} \to 2 \), so the expression becomes: \[ \frac{1}{2 + 2} = \frac{1}{4}. \]

Step 5: Conclusion.
The solution to the limit is \( \boxed{\frac{1}{4}} \).