We are given the function:

$f(x) = \frac{|x - 10|}{x - 10}$

This function can be interpreted based on the behavior of $x$ relative to 10:

• When $x > 10$, $|x - 10| = x - 10$, so $f(x) = \frac{x - 10}{x - 10} = 1$.
• When $x < 10$, $|x - 10| = -(x - 10)$, so $f(x) = \frac{-(x - 10)}{x - 10} = -1$.
• When $x = 10$, the denominator becomes 0, making $f(x)$ undefined at $x = 10$.

Now, let's fill out the table based on this understanding:

$x$	9.9	9.99	9.999	10.001	10.01	10.1
$f(x)$	-1	-1	-1	1	1	1

Estimating the limit:

As $x \to 10$ from the left (i.e., for values of $x$ like 9.9, 9.99, and 9.999), $f(x) = -1$.

As $x \to 10$ from the right (i.e., for values of $x$ like 10.001, 10.01, and 10.1), $f(x) = 1$.

Since the left-hand limit ($\lim_{x \to 10^-} f(x) = -1$) and the right-hand limit ($\lim_{x \to 10^+} f(x) = 1$) are not equal, the overall limit does not exist.

Thus, the limit is:

$\lim_{x \to 10} f(x) = \text{DNE} \quad \text{(Does Not Exist)}$

Would you like further explanation or have any questions? Here are some related questions to explore:

1. What is the definition of a limit?
2. How do you determine one-sided limits?
3. Can a limit exist if the function behaves differently from both sides of the point?
4. What does it mean when a function is undefined at a point?
5. How can we graphically interpret the behavior of $f(x)$ as $x \to 10$?

Tip: One-sided limits can provide valuable insight into a function's behavior, even when the overall limit does not exist.