omplete the following table and use the results to estimate the following limit:
8𝑥
x 𝑓(𝑥) =
√2𝑥 + 16 − 4
-0.5
8𝑥
-0.1
lim
𝑥→0
≈
√2𝑥+16−4
_____________
-0.01
0.01
0.1
0.5

We are tasked with estimating the following limit:

\[
\lim_{{x \to 0}} \frac{8x}{\sqrt{2x + 16} - 4}
\]

To estimate the limit, we will evaluate the expression for various small values of \(x\), filling in a table. The function to compute is:

\[
f(x) = \frac{8x}{\sqrt{2x + 16} - 4}
\]

### Steps:
1. For each \(x\) value, substitute into the function and compute \(f(x)\).
2. Create a table for the given \(x\) values: -0.5, -0.1, -0.01, 0.01, 0.1, and 0.5.
3. Observe the trend as \(x\) approaches 0 from both the negative and positive sides to estimate the limit.

Let me calculate the values for the table.Here are the computed values for \( f(x) = \frac{8x}{\sqrt{2x + 16} - 4} \) at different \(x\) values:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-0.5 & 31.492 \\
-0.1 & 31.900 \\
-0.01 & 31.990 \\
0.01 & 32.010 \\
0.1 & 32.100 \\
0.5 & 32.492 \\
\hline
\end{array}
\]

### Estimation of the Limit:
As \(x \to 0\), the values of \(f(x)\) are approaching approximately **32**. Therefore, we estimate:

\[
\lim_{{x \to 0}} \frac{8x}{\sqrt{2x + 16} - 4} \approx 32
\]

Would you like more details or further clarification on this result?

---

Here are five follow-up questions you might explore:
1. How is the limit evaluated using algebraic techniques such as rationalization?
2. How does the behavior of \(f(x)\) change for values of \(x\) further away from zero?
3. Can we confirm this limit using L'Hopital's Rule?
4. How would the limit change if the function involved higher powers of \(x\)?
5. What is the geometric interpretation of the square root function in this problem?

**Tip:** When estimating limits numerically, check values from both the positive and negative sides to ensure consistency.

Complete the following table and use the results to estimate the following limit: lim (x → 0) 8x / (√(2x + 16) - 4)

Learn how to estimate the limit lim (x → 0) 8x / (√(2x + 16) - 4) by filling out a table of values and observing trends. This problem covers key concepts in limits, algebra, and numerical estimation, suitable for Grades 10-12.

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