If f(x) = `(sqrt(4 +x)-2)/(x)` is continuous x=0 then the value of f(0) is

Options

A. 0
B. `(1)/(2)`
C. `(1)/(4)`
D. not defined

Answer

To determine the value of \( f(0) \) for the function \( f(x) = \frac{\sqrt{4 + x} - 2}{x} \) such that it is continuous at \( x = 0 \), we need to evaluate the limit of \( f(x) \) as \( x \) approaches 0.

### Step-by-Step Solution:

1. **Identify the Function**:
\[ f(x) = \frac{\sqrt{4 + x} - 2}{x} \]

2. **Check for Direct Substitution**:
When we substitute \( x = 0 \):
\[ f(0) = \frac{\sqrt{4 + 0} - 2}{0} = \frac{2 - 2}{0} = \frac{0}{0} \]
This is an indeterminate form, so we need to simplify the expression.

3. **Rationalize the Numerator**:
To simplify \( f(x) \), we can multiply the numerator and the denominator by the conjugate of the numerator:
\[ f(x) = \frac{\sqrt{4 + x} - 2}{x} \cdot \frac{\sqrt{4 + x} + 2}{\sqrt{4 + x} + 2} \]
This gives:
\[ f(x) = \frac{(\sqrt{4 + x} - 2)(\sqrt{4 + x} + 2)}{x(\sqrt{4 + x} + 2)} = \frac{(4 + x) - 4}{x(\sqrt{4 + x} + 2)} \]
Simplifying the numerator:
\[ f(x) = \frac{x}{x(\sqrt{4 + x} + 2)} \]

4. **Cancel Common Terms**:
We can cancel \( x \) in the numerator and denominator (for \( x \neq 0 \)):
\[ f(x) = \frac{1}{\sqrt{4 + x} + 2} \]

5. **Evaluate the Limit as \( x \) Approaches 0**:
Now we find the limit as \( x \) approaches 0:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{1}{\sqrt{4 + x} + 2} \]
Substituting \( x = 0 \):
\[ = \frac{1}{\sqrt{4 + 0} + 2} = \frac{1}{2 + 2} = \frac{1}{4} \]

6. **Conclusion**:
Since \( f(x) \) is continuous at \( x = 0 \), we can define \( f(0) \) as:
\[ f(0) = \frac{1}{4} \]

### Final Answer:
\[ f(0) = \frac{1}{4} \]