Consider the function :
`f(x)={{:(x+2",", x ne 1), (0",", x =1.):}`
To find `lim_(x to 1)f(x)`.

Options

A. 1
B. 2
C. does not exist
D. none of these

Answer

To find the limit of the function \( f(x) \) as \( x \) approaches 1, we will analyze the function given:

\[ f(x) = \begin{cases} x + 2 & \text{if } x \neq 1 \\ 0 & \text{if } x = 1 \end{cases} \]

### Step-by-Step Solution:

1. **Identify the limit we need to find:**
We need to evaluate \( \lim_{x \to 1} f(x) \).

2. **Consider the definition of the function:**
Since we are looking for the limit as \( x \) approaches 1, we will use the part of the function that applies when \( x \) is not equal to 1. Thus, we will use \( f(x) = x + 2 \) for our calculations.

3. **Substitute \( x \) with values approaching 1:**
We can directly substitute \( x = 1 \) into the expression \( x + 2 \):
\[ f(x) = x + 2 \implies f(1) = 1 + 2 = 3 \]

4. **Conclude the limit:**
Therefore, we find that:
\[ \lim_{x \to 1} f(x) = 3 \]

### Final Answer:
\[ \lim_{x \to 1} f(x) = 3 \]