Consider the function `f(x)={{:(x^(2)-5,xle3),(sqrt(x+13),xgt3):}`
What is `lim_(xto3)f(x)` equal to ?

Options

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

Explanation

To find the limit of the function \( f(x) \) as \( x \) approaches 3, we need to analyze the function given in the problem. The function is defined piecewise:

\[ f(x) = \begin{cases} x^2 - 5 & \text{if } x \leq 3 \\ \sqrt{x + 13} & \text{if } x > 3 \end{cases} \]

We are interested in calculating \( \lim_{x \to 3} f(x) \).

### Step 1: Determine the left-hand limit as \( x \) approaches 3

Since we are approaching 3 from the left (values less than or equal to 3), we will use the first part of the piecewise function:

\[ f(x) = x^2 - 5 \quad \text{for } x \leq 3 \]

Now we calculate the limit:

\[ \lim_{x \to 3^-} f(x) = \lim_{x \to 3} (x^2 - 5) \]

Substituting \( x = 3 \):

\[ = 3^2 - 5 = 9 - 5 = 4 \]

### Step 2: Determine the right-hand limit as \( x \) approaches 3

Now we will calculate the limit as \( x \) approaches 3 from the right (values greater than 3):

\[ f(x) = \sqrt{x + 13} \quad \text{for } x > 3 \]

Now we calculate the limit:

\[ \lim_{x \to 3^+} f(x) = \lim_{x \to 3} \sqrt{x + 13} \]

Substituting \( x = 3 \):

\[ = \sqrt{3 + 13} = \sqrt{16} = 4 \]

### Step 3: Combine the results

Since both the left-hand limit and the right-hand limit as \( x \) approaches 3 are equal:

\[ \lim_{x \to 3^-} f(x) = 4 \quad \text{and} \quad \lim_{x \to 3^+} f(x) = 4 \]

Thus, we can conclude that:

\[ \lim_{x \to 3} f(x) = 4 \]

### Final Answer

\[ \lim_{x \to 3} f(x) = 4 \]