A function f(x) is defined as f(x)= `{:{(1," when " x != 0),(2," when " x = 0 ):}` does the Limit of f(x) as `x to 0 ` exist ? Explain your answer .

Answer

To determine whether the limit of the function \( f(x) \) as \( x \) approaches 0 exists, we need to analyze the function and calculate the left-hand limit and the right-hand limit.

### Step-by-Step Solution:

1. **Define the Function**:
The function \( f(x) \) is defined as:
\[ f(x) = \begin{cases} 1 & \text{when } x \neq 0 \\ 2 & \text{when } x = 0 \end{cases} \]

2. **Calculate the Right-Hand Limit**:
The right-hand limit as \( x \) approaches 0 is given by:
\[ \lim_{x \to 0^+} f(x) \]
Since \( f(x) = 1 \) when \( x \neq 0 \), we can write:
\[ \lim_{x \to 0^+} f(x) = 1 \]

3. **Calculate the Left-Hand Limit**:
The left-hand limit as \( x \) approaches 0 is given by:
\[ \lim_{x \to 0^-} f(x) \]
Similarly, since \( f(x) = 1 \) when \( x \neq 0 \), we have:
\[ \lim_{x \to 0^-} f(x) = 1 \]

4. **Compare the Limits**:
Now we compare the right-hand limit and the left-hand limit:
\[ \lim_{x \to 0^+} f(x) = 1 \quad \text{and} \quad \lim_{x \to 0^-} f(x) = 1 \]
Since both limits are equal, we can conclude:
\[ \lim_{x \to 0} f(x) = 1 \]

5. **Conclusion**:
The limit of \( f(x) \) as \( x \) approaches 0 exists and is equal to 1. However, it is important to note that \( f(0) = 2 \), which means the function does not equal the limit at that point.

### Final Answer:
The limit of \( f(x) \) as \( x \to 0 \) exists and is equal to 1, but \( f(0) \) is 2.

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