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:

\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.