The function you have is:

$f(x) = \frac{x^2 - 4x}{x^2 - 16}$

To begin, simplify the function and then calculate the values for $f(x)$ at the given $x$ values.

Step 1: Simplifying the function

$f(x) = \frac{x^2 - 4x}{x^2 - 16}$

Notice that $x^2 - 16$ can be factored as a difference of squares:

$x^2 - 16 = (x - 4)(x + 4)$

Thus, the function becomes:

$f(x) = \frac{x(x - 4)}{(x - 4)(x + 4)}$

For all $x \neq 4$, the $(x - 4)$ terms cancel out, leaving:

$f(x) = \frac{x}{x + 4}, \quad \text{for} \quad x \neq 4$

At $x = 4$, the original form results in division by zero, so the function is undefined there.

Step 2: Calculate $f(x)$ for the given values of $x$

For each of the following values, we'll compute $f(x) = \frac{x}{x + 4}$.

1. $x = 4.1$:

$f(4.1) = \frac{4.1}{4.1 + 4} = \frac{4.1}{8.1} \approx 0.50617$

2. $x = 4.05$:

$f(4.05) = \frac{4.05}{4.05 + 4} = \frac{4.05}{8.05} \approx 0.50311$

3. $x = 4.01$:

$f(4.01) = \frac{4.01}{4.01 + 4} = \frac{4.01}{8.01} \approx 0.50062$

4. $x = 4.001$:

$f(4.001) = \frac{4.001}{4.001 + 4} = \frac{4.001}{8.001} \approx 0.50012$

5. $x = 4.0001$:

$f(4.0001) = \frac{4.0001}{4.0001 + 4} = \frac{4.0001}{8.0001} \approx 0.50001$

Now, for values of $x$ approaching from the left (less than 4):

6. $x = 3.9$:

$f(3.9) = \frac{3.9}{3.9 + 4} = \frac{3.9}{7.9} \approx 0.49367$

7. $x = 3.95$:

$f(3.95) = \frac{3.95}{3.95 + 4} = \frac{3.95}{7.95} \approx 0.49748$

8. $x = 3.99$:

$f(3.99) = \frac{3.99}{3.99 + 4} = \frac{3.99}{7.99} \approx 0.49937$

9. $x = 3.999$:

$f(3.999) = \frac{3.999}{3.999 + 4} = \frac{3.999}{7.999} \approx 0.49987$

10. $x = 3.9999$:

$f(3.9999) = \frac{3.9999}{3.9999 + 4} = \frac{3.9999}{7.9999} \approx 0.49999$

Summary of Results

$x$	$f(x)$
4.1	0.50617
4.05	0.50311
4.01	0.50062
4.001	0.50012
4.0001	0.50001
3.9	0.49367
3.95	0.49748
3.99	0.49937
3.999	0.49987
3.9999	0.49999

Do you need further details or clarification? Here are some related questions you might want to explore:

1. What happens at $x = 4$? Why is it undefined?
2. How does the behavior of the function change as $x$ approaches 4 from the left and right?
3. Can we graph the function and identify key features such as asymptotes?
4. What is the limit of $f(x)$ as $x \to 4$?
5. Can this function be further generalized for other values of parameters?

Tip: When working with rational functions, always check for points of discontinuity by factoring and simplifying the expressions.