If `F (x) = {{:(-x^(2) + 1, x lt 0),(0,x = 0),(x^(2) + 1,x gt = 0):}`, .then `lim_(x to 0) f(x)` is

Options

A. 0
B. 1
C. 2
D. `-1`

Answer

To find the limit of the function \( F(x) \) as \( x \) approaches 0, we need to evaluate the left-hand limit and the right-hand limit.

### Step-by-Step Solution:

1. **Identify the function pieces**:
The function \( F(x) \) is defined as:
- \( F(x) = -x^2 + 1 \) for \( x < 0 \)
- \( F(x) = 0 \) for \( x = 0 \)
- \( F(x) = x^2 + 1 \) for \( x > 0 \)

2. **Calculate the left-hand limit**:
We need to find \( \lim_{x \to 0^-} F(x) \):
- Since \( x \) is approaching 0 from the left (negative side), we use the piece of the function defined for \( x < 0 \):
\[ F(x) = -x^2 + 1 \]
- Now, substitute \( x = 0 \) into this expression:
\[ \lim_{x \to 0^-} F(x) = -0^2 + 1 = 1 \]

3. **Calculate the right-hand limit**:
Now we find \( \lim_{x \to 0^+} F(x) \):
- Since \( x \) is approaching 0 from the right (positive side), we use the piece of the function defined for \( x > 0 \):
\[ F(x) = x^2 + 1 \]
- Substitute \( x = 0 \) into this expression:
\[ \lim_{x \to 0^+} F(x) = 0^2 + 1 = 1 \]

4. **Compare the limits**:
- We found that:
\[ \lim_{x \to 0^-} F(x) = 1 \]
\[ \lim_{x \to 0^+} F(x) = 1 \]
- Since both the left-hand limit and the right-hand limit are equal, we can conclude that:
\[ \lim_{x \to 0} F(x) = 1 \]

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