Use a table of values to estimate the value of the limit (correct to three decimal places). If you have a graphing device, use it to confirm your result graphically.

lim *t*→0 

9*t* − 1

*t*


To estimate the limit of the function \(\lim_{t \to 0} \frac{9t - 1}{t}\), we can start by creating a table of values for the function \(\frac{9t - 1}{t}\) as \(t\) approaches 0 from both the left (\(t \to 0^-\)) and the right (\(t \to 0^+\)).

Let's calculate the values:

\[
f(t) = \frac{9t - 1}{t} = 9 - \frac{1}{t}
\]

Now, we'll choose values for \(t\) close to 0:

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

### Analysis

- As \(t\) approaches 0 from the left (\(t \to 0^-\)), \(f(t)\) becomes increasingly negative, approaching \(-\infty\).
- As \(t\) approaches 0 from the right (\(t \to 0^+\)), \(f(t)\) becomes increasingly positive, approaching \(+\infty\).

### Conclusion

Since the function diverges in different directions depending on the side from which \(t\) approaches 0, the limit \(\lim_{t \to 0} \frac{9t - 1}{t}\) **does not exist**. The function has a vertical asymptote at \(t = 0\).

#### Graphical Confirmation

If you use a graphing device, you should observe the same behavior: the function shoots downward as \(t\) approaches 0 from the left and upward as \(t\) approaches 0 from the right, confirming that the limit does not exist.

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

### Related Questions:
1. What are one-sided limits, and how are they used to evaluate limits?
2. How do you identify asymptotes in a function's graph?
3. What is the difference between a limit existing and a limit approaching infinity?
4. Can you provide an example where the limit does exist as \(t\) approaches 0?
5. How do piecewise functions impact limit calculations at a point?

### Tip:
When estimating limits numerically or graphically, always consider approaching the point of interest from both directions to determine if the limit is the same or diverges.

Learn how to estimate the limit of (9t - 1) / t as t approaches 0 using numerical methods and graphical confirmation. Understand why the limit does not exist due to divergent behavior in different directions. Suitable for advanced high school students and anyone interested in limit analysis in calculus.

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