Then answer this question. Use a table of values to estimate the value of the limit (correct to three decimal places).

Sure! Let's use a table of values to estimate the value of the following limit:

\[
\lim_{t \to 0} \frac{9t - 1}{t}
\]

We will choose values of \(t\) close to 0 from both the left (\(t \to 0^-\)) and the right (\(t \to 0^+\)), and compute the corresponding values of the function \(\frac{9t - 1}{t}\).

### Table of Values

| \(t\)     | \(f(t) = \frac{9t - 1}{t}\) |
|-----------|-----------------------------|
| \(-0.1\)  | \(-19\)                      |
| \(-0.01\) | \(-91\)                      |
| \(-0.001\)| \(-991\)                     |
| \(-0.0001\)| \(-9991\)                   |
| \(0.0001\) | \(8991\)                    |
| \(0.001\) | \(8991\)                     |
| \(0.01\)  | \(891\)                      |
| \(0.1\)   | \(89\)                       |

### Analysis

- As \(t\) approaches 0 from the left (\(t \to 0^-\)), the function values become very large in the negative direction.
- As \(t\) approaches 0 from the right (\(t \to 0^+\)), the function values become very large in the positive direction.

### Estimation of the Limit

The function does not approach a single finite value as \(t\) approaches 0. Instead, it tends to \(-\infty\) from the left and \(+\infty\) from the right.

### Conclusion

Since the function diverges in opposite directions depending on whether \(t\) approaches 0 from the left or the right, the limit:

\[
\lim_{t \to 0} \frac{9t - 1}{t}
\]

**does not exist**.

Would you like any further details or have any other questions?

### Related Questions:
1. What happens to the limit if the function was \(\frac{t - 1}{t}\)?
2. How does the concept of limits apply to continuous functions?
3. What are the criteria for a limit to exist at a point?
4. Can the function be modified to make the limit exist at \(t = 0\)?
5. What are practical scenarios where understanding limits is crucial?

### Tip:
When analyzing limits, always check the behavior of the function from both sides of the point of interest to determine if the limit exists and if it's finite.

Learn how to estimate the limit of (9t - 1) / t using a table of values approach. Understand the behavior of the function as t approaches 0 from both sides and why the limit does not exist. Suitable for advanced high school students studying limits in calculus.

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