Read the following information carefully and answer the question given below:
Consider the function `f(x)={{:(x^(2)-5,xle3),(sqrt(x+13),xgt3):}`
What is `underset(xrarr3)(lim)f(x)` equal to:

Options

A. 2
B. 4
C. 5
D. 13

Answer

To find the limit of the function \( f(x) \) as \( x \) approaches 3, we will evaluate both the left-hand limit (LHL) and the right-hand limit (RHL) at that point.

### Step-by-Step Solution:

1. **Identify the Function**:
The function is defined as:
\[ f(x) = \begin{cases} x^2 - 5 & \text{if } x \leq 3 \\ \sqrt{x + 13} & \text{if } x > 3 \end{cases} \]

2. **Calculate the Left-Hand Limit (LHL)**:
We need to find \( \lim_{x \to 3^-} f(x) \). Since we are approaching from the left (where \( x \leq 3 \)), we use the first part of the function:
\[ f(x) = x^2 - 5 \]
Now, substituting \( x = 3 \):
\[ \lim_{x \to 3^-} f(x) = 3^2 - 5 = 9 - 5 = 4 \]

3. **Calculate the Right-Hand Limit (RHL)**:
Next, we find \( \lim_{x \to 3^+} f(x) \). Since we are approaching from the right (where \( x > 3 \)), we use the second part of the function:
\[ f(x) = \sqrt{x + 13} \]
Now, substituting \( x = 3 \):
\[ \lim_{x \to 3^+} f(x) = \sqrt{3 + 13} = \sqrt{16} = 4 \]

4. **Compare LHL and RHL**:
We have found:
\[ \lim_{x \to 3^-} f(x) = 4 \quad \text{and} \quad \lim_{x \to 3^+} f(x) = 4 \]
Since both limits are equal, we conclude that:
\[ \lim_{x \to 3} f(x) = 4 \]

5. **Final Result**:
Therefore, the limit of \( f(x) \) as \( x \) approaches 3 is:
\[ \lim_{x \to 3} f(x) = 4 \]