Find \( \lim _{x \rightarrow 14} f(x) \) using the table shown below. Be sure to fill out the table and round all numbers to three decimal places.\[
f(x)=\frac{x-14}{\sqrt{x}-\sqrt{14}}
\]\[\begin{array}{|c|c|c|c|c|c|c|c|}
\hline\text{ x }&\text{ 13.9 }&\text{ 13.99 }&\text{ 13.999 }&\text{ 14 }&\text{ 14.001 }&\text{ 14.01 }&\text{ 14.1 }\\
\hline f(x)  & & & &\text{ X }& & & \\
\hline
\end{array}\]\[
\lim _{x \rightarrow 14} f(x)=
\]\( \square \)

To find the limit \( \lim _{x \rightarrow 14} f(x) \) using the table, we need to evaluate the given function:

\[
f(x) = \frac{x - 14}{\sqrt{x} - \sqrt{14}}
\]

### Step 1: Simplify the function
The expression involves a difference of square roots, so let's rationalize the denominator by multiplying the numerator and denominator by \( \sqrt{x} + \sqrt{14} \):

\[
f(x) = \frac{(x - 14)(\sqrt{x} + \sqrt{14})}{(\sqrt{x} - \sqrt{14})(\sqrt{x} + \sqrt{14})}
\]

Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \), the denominator simplifies:

\[
f(x) = \frac{(x - 14)(\sqrt{x} + \sqrt{14})}{x - 14}
\]

For \( x \neq 14 \), the \( x - 14 \) terms cancel out:

\[
f(x) = \sqrt{x} + \sqrt{14}
\]

### Step 2: Evaluate the function at different values of \( x \)
Now, we can compute the values of \( f(x) \) for \( x \) values close to 14:

\[
f(x) = \sqrt{x} + \sqrt{14}
\]

- For \( x = 13.9 \): \( f(13.9) = \sqrt{13.9} + \sqrt{14} = 3.728 + 3.742 = 7.470 \)
- For \( x = 13.99 \): \( f(13.99) = \sqrt{13.99} + \sqrt{14} = 3.739 + 3.742 = 7.481 \)
- For \( x = 13.999 \): \( f(13.999) = \sqrt{13.999} + \sqrt{14} = 3.741 + 3.742 = 7.483 \)
- For \( x = 14 \): \( f(14) = \sqrt{14} + \sqrt{14} = 3.742 + 3.742 = 7.484 \)
- For \( x = 14.001 \): \( f(14.001) = \sqrt{14.001} + \sqrt{14} = 3.743 + 3.742 = 7.485 \)
- For \( x = 14.01 \): \( f(14.01) = \sqrt{14.01} + \sqrt{14} = 3.744 + 3.742 = 7.486 \)
- For \( x = 14.1 \): \( f(14.1) = \sqrt{14.1} + \sqrt{14} = 3.753 + 3.742 = 7.495 \)

### Step 3: Fill in the table
\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline\text{ x }&\text{ 13.9 }&\text{ 13.99 }&\text{ 13.999 }&\text{ 14 }&\text{ 14.001 }&\text{ 14.01 }&\text{ 14.1 }\\
\hline f(x)  & 7.470 & 7.481 & 7.483 & 7.484 & 7.485 & 7.486 & 7.495 \\
\hline
\end{array}
\]

### Step 4: Determine the limit
From the table, as \( x \) approaches 14 from both sides, the values of \( f(x) \) approach approximately 7.484. Therefore, the limit is:

\[
\lim _{x \rightarrow 14} f(x) = 7.484
\]

Let me know if you have any questions or if you'd like more details!

Here are 5 related questions:
1. How would you rationalize a more complex expression with square roots?
2. What are other methods to find limits besides using a table?
3. Can this limit be found using L'Hopital's Rule?
4. What is the difference between a one-sided limit and a two-sided limit?
5. How does the concept of continuity relate to finding limits?

**Tip:** Rationalizing the denominator is a useful trick when dealing with square roots in limits!

Find \( \lim _{x \rightarrow 14} f(x) \) using the table shown below. Be sure to fill out the table and round all numbers to three decimal places.
\[ f(x)=\frac{x-14}{\sqrt{x}-\sqrt{14}} \]
\[\begin{array}{|c|c|c|c|c|c|c|c|} \hline\text{ x }&\text{ 13.9 }&\text{ 13.99 }&\text{ 13.999 }&\text{ 14 }&\text{ 14.001 }&\text{ 14.01 }&\text{ 14.1 }\\ \hline f(x)  & & & &\text{ X }& & & \\ \hline \end{array}\]
\( \lim _{x \rightarrow 14} f(x)= \square \)

Solve the limit \( \lim _{x \rightarrow 14} \frac{x-14}{\sqrt{x}-\sqrt{14}} \) by filling in a table and using rationalization techniques. This problem involves limits and rationalizing square roots, perfect for students in Grades 11-12. The limit evaluates to approximately 7.484.

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