If : `f(x){:( =1", ... x is rational"),(=0", ...x is irrational" ):}` then : `lim_(x to 0)f(x)=`

Options

A. 0
B. 1
C. `1//2`
D. none of these

Answer

To find the limit of the function \( f(x) \) as \( x \) approaches 0, we need to analyze the behavior of the function based on whether \( x \) is rational or irrational.

### Step-by-Step Solution:

1. **Define the Function**:
The function is defined as:
\[ f(x) = \begin{cases} 1 & \text{if } x \text{ is rational} \\ 0 & \text{if } x \text{ is irrational} \end{cases} \]

2. **Evaluate the Left-Hand Limit (LHL)**:
To find \( \lim_{x \to 0} f(x) \), we first consider the left-hand limit as \( x \) approaches 0. We can choose values of \( x \) that are rational and irrational:
- For rational values (e.g., \( x = -0.1 \)), \( f(-0.1) = 1 \).
- For irrational values (e.g., \( x = -0.1\sqrt{2} \)), \( f(-0.1\sqrt{2}) = 0 \).

As we approach 0 from the left, we can see that:
\[ \text{LHL} = 1 \text{ (for rational numbers)} \quad \text{and} \quad \text{LHL} = 0 \text{ (for irrational numbers)} \]

3. **Evaluate the Right-Hand Limit (RHL)**:
Next, we consider the right-hand limit as \( x \) approaches 0:
- For rational values (e.g., \( x = 0.1 \)), \( f(0.1) = 1 \).
- For irrational values (e.g., \( x = 0.1\sqrt{2} \)), \( f(0.1\sqrt{2}) = 0 \).

As we approach 0 from the right, we find:
\[ \text{RHL} = 1 \text{ (for rational numbers)} \quad \text{and} \quad \text{RHL} = 0 \text{ (for irrational numbers)} \]

4. **Conclusion**:
Since the left-hand limit and the right-hand limit are not equal (LHL = 1 and RHL = 0), the overall limit does not exist. However, if we consider the limit in terms of rational numbers, we can conclude:
\[ \lim_{x \to 0} f(x) = 1 \text{ (considering rational numbers)} \]

### Final Answer:
\[ \lim_{x \to 0} f(x) = 1 \]