To solve this problem, let's address it step-by-step.

(a) Completing the table

The function given is: $f(x) = \frac{x+2}{\sqrt{2 - x} - 2}$

We'll calculate the values of $f(x)$ for the given $x$ values in the table, rounding to four decimal places.

Calculations

1. For $x = -2.1$:

$f(-2.1) = \frac{-2.1 + 2}{\sqrt{2 - (-2.1)} - 2} = \frac{-0.1}{\sqrt{4.1} - 2}$

2. For $x = -2.01$:

$f(-2.01) = \frac{-2.01 + 2}{\sqrt{2 - (-2.01)} - 2} = \frac{-0.01}{\sqrt{4.01} - 2}$

3. For $x = -2.001$:

$f(-2.001) = \frac{-2.001 + 2}{\sqrt{2 - (-2.001)} - 2} = \frac{-0.001}{\sqrt{4.001} - 2}$

4. For $x = -1.999$:

$f(-1.999) = \frac{-1.999 + 2}{\sqrt{2 - (-1.999)} - 2} = \frac{0.001}{\sqrt{3.999} - 2}$

5. For $x = -1.99$:

$f(-1.99) = \frac{-1.99 + 2}{\sqrt{2 - (-1.99)} - 2} = \frac{0.01}{\sqrt{3.99} - 2}$

6. For $x = -1.9$:

$f(-1.9) = \frac{-1.9 + 2}{\sqrt{2 - (-1.9)} - 2} = \frac{0.1}{\sqrt{3.9} - 2}$

I will calculate each value now. math

Define the function f(x)

def f(x): numerator = x + 2 denominator = math.sqrt(2 - x) - 2 return numerator / denominator

List of x values

x_values = [-2.1, -2.01, -2.001, -1.999, -1.99, -1.9]

Calculate f(x) for each x value

results = {x: round(f(x), 4) for x in x_values}

resultsHere are the computed values of $f(x)$ for each $x$:

(b) Estimating the Limit

To estimate: $\lim_{x \to -2} \frac{x+2}{\sqrt{2-x} - 2}$

From the table, as $x \to -2$, $f(x)$ appears to approach $-4$. Hence, the estimated limit is: $\boxed{-4}$

Do you want further clarification or additional steps?

Related Questions

1. What happens if the denominator in $f(x)$ is zero?
2. How do we confirm limits using a graph?
3. Why does $f(x)$ behave the way it does near $x = -2$?
4. Can the function be simplified to avoid division by zero?
5. What if $x$ values were positive? How would $f(x)$ behave?

Tip

Always verify limits numerically and analytically for accuracy.