Step 1: Continuity Condition
For continuity at \( x = 0 \), the limit of \( f(x) \) as \( x \) approaches 0 must equal \( f(0) \), which is \( k \). So, \( \lim_{x \to 0} f(x) = f(0) = k \).
Step 2: Limit Evaluation
For \( x eq 0 \), the function is given by:
\[
f(x) = \frac{\sqrt{4} + x - 2x}{x}.
\]
To evaluate the limit, multiply the numerator and denominator by the conjugate of the numerator's non-radical term, which is \( \sqrt{4} + x + 2 \):
\[
f(x) = \frac{(\sqrt{4} + x - 2)(\sqrt{4} + x + 2)}{x(\sqrt{4} + x + 2)}
\]
This simplifies to:
\[
= \frac{(4 + x) - 4}{x(\sqrt{4} + x + 2)}
\]
\[
= \frac{x}{x(\sqrt{4} + x + 2)}
\]
For \( x eq 0 \), we can cancel \( x \):
\[
= \frac{1}{\sqrt{4} + x + 2}.
\]
Step 3: Substitution at \( x = 0 \)
Now, substitute \( x = 0 \) into the simplified expression to find the limit:
\[
\lim_{x \to 0} f(x) = \frac{1}{\sqrt{4} + 0 + 2} = \frac{1}{2 + 0 + 2} = \frac{1}{4}.
\]
Step 4: Determining \( k \)
From Step 1, we know that \( k = \lim_{x \to 0} f(x) \). Therefore, \( k = \frac{1}{4} \). This matches option (B).