Complete the table by computing f(x) at the given values of x.
f(x) =
|x − 10|
x − 10

x	9.9	9.99	9.999	10.001	10.01	10.1
f(x)
Use these results to estimate the indicated limit (if it exists). (If an answer does not exist, enter DNE.)
lim
x → 10
f(x)

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.

Complete the table by computing f(x) at the given values of x.
f(x) = |x − 10| / (x − 10)

x: 9.9, 9.99, 9.999, 10.001, 10.01, 10.1

Use these results to estimate the indicated limit (if it exists).
lim (x → 10) f(x)

Learn how to compute the limit of f(x) = |x − 10| / (x − 10) as x approaches 10. This problem explores piecewise functions and one-sided limits, demonstrating how the function behaves differently from the left and right. Suitable for students in Grades 11-12.

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