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.