Consider the function :
`" "f(x)={{:(x-2",", x lt 0), (" 0,", x=0), (x+2",", x gt 0.):}`
To find `lim_(x to 0) f(x)`.

Options

A. -2
B. 2
C. limit does not exist
D. none of these

Answer

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

### Step 1: Define the function
The function \( f(x) \) is defined as follows:
- \( f(x) = x - 2 \) for \( x < 0 \)
- \( f(x) = 0 \) for \( x = 0 \)
- \( f(x) = x + 2 \) for \( x > 0 \)

### Step 2: Find the left-hand limit
To find the left-hand limit as \( x \) approaches 0, we calculate:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x - 2) \]
As \( x \) approaches 0 from the left (values less than 0), we substitute \( x = 0 \):
\[ = 0 - 2 = -2 \]

### Step 3: Find the right-hand limit
Next, we find the right-hand limit as \( x \) approaches 0:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x + 2) \]
As \( x \) approaches 0 from the right (values greater than 0), we substitute \( x = 0 \):
\[ = 0 + 2 = 2 \]

### Step 4: Compare the limits
Now we compare the left-hand limit and the right-hand limit:
- Left-hand limit: \( -2 \)
- Right-hand limit: \( 2 \)

Since the left-hand limit is not equal to the right-hand limit, we conclude that the limit does not exist.

### Final Answer
\[ \lim_{x \to 0} f(x) \text{ does not exist.} \]

---