Let's break this problem into two parts:

Part (a): Completing the Table

We are given the function $f(x) = \frac{x + 13 - \sqrt{-x}}{2}$ and asked to compute the values of $f(x)$ for specific $x$-values. We will plug in the values of $x$ into the function and calculate them, rounding the results to four decimal places.

The values of $x$ to compute are: $-1.1$, $-1.01$, $-1.001$, $-0.999$, $-0.99$, and $-0.9$.

I'll compute each value and fill the table.

$f(x) = \frac{x + 13 - \sqrt{-x}}{2}$

Let me compute these values now.Here is the completed table with the calculated values of $f(x)$:

\hline x & f(x) \\ \hline -1.1 & 5.4256 \\ -1.01 & 5.4925 \\ -1.001 & 5.4993 \\ -0.999 & 5.5008 \\ -0.99 & 5.5075 \\ -0.9 & 5.5757 \\ \hline \end{array}$$ ### Part (b): Estimating $$\lim_{x \to -1} \left( \frac{x + 13 - \sqrt{-x}}{2} \right)$$ From the table, as $$x$$ approaches $$-1$$, the values of $$f(x)$$ appear to be approaching approximately $$5.5$$. Therefore, we can estimate: $$\lim_{x \to -1} \left( \frac{x + 13 - \sqrt{-x}}{2} \right) \approx 5.5$$ Would you like more details or have any questions? Here are 5 related questions to consider: 1. What happens to the function as $$x \to 0$$? 2. Can the function be simplified for larger $$x$$-values? 3. How does the square root of negative $$x$$ affect the function’s behavior? 4. What would happen if $$x$$ approached positive values close to zero? 5. Can you find the derivative of $$f(x)$$? **Tip:** When estimating limits numerically, look at the trend of the function's output as the input values approach the limit.