(a) Complete the table below given  f(x)=x+13−x−−−−√−2
 .

 

Round your answers to four decimal places.

 

x
 	 −1.1
 	 −1.01
 	 −1.001
 	 −0.999
 	 −0.99
 	 −0.9
 
f(x)
 	
Number
Number
Number
Number
Number
Number
 

(b) Estimate  limx→−1(x+13−x−−−−√−2)

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 \( f(x) \):

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-1.1   & 5.4256 \\
-1.01  & 5.4925 \\
-1.001 & 5.4993 \\
-0.999 & 5.5008 \\
-0.99  & 5.5075 \\
-0.9   & 5.5757 \\
\hline
\end{array}
\]

### 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.

(a) Complete the table below given f(x) = (x + 13 - sqrt(-x))/2 . Round your answers to four decimal places. x: -1.1, -1.01, -1.001, -0.999, -0.99, -0.9. f(x): Number, Number, Number, Number, Number, Number. (b) Estimate lim(x→-1) (x + 13 - sqrt(-x))/2.

Understand how to compute the values of the function f(x) = (x + 13 - sqrt(-x))/2 for specific x-values and estimate the limit as x approaches -1. This problem involves analyzing square roots and applying the Limit Theorem. Suitable for students in Grades 11-12.

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