To estimate the limit $\lim_{{t \to 0}} \frac{6t - 1}{t}$, we can approach it by creating a table of values for $t$ approaching 0 from both the positive and negative sides. This will help us estimate the behavior of the function as $t$ gets closer to 0.

The expression can be simplified to: $\frac{6t - 1}{t} = \frac{6t}{t} - \frac{1}{t} = 6 - \frac{1}{t}$

Now, let's create a table for values of $t$ approaching 0:

$t$	$\frac{6t - 1}{t}$
0.1	$6 - \frac{1}{0.1} = 6 - 10 = -4$
0.01	$6 - \frac{1}{0.01} = 6 - 100 = -94$
0.001	$6 - \frac{1}{0.001} = 6 - 1000 = -994$
-0.1	$6 - \frac{1}{-0.1} = 6 + 10 = 16$
-0.01	$6 - \frac{1}{-0.01} = 6 + 100 = 106$
-0.001	$6 - \frac{1}{-0.001} = 6 + 1000 = 1006$

From this table, as $t$ approaches 0 from the positive side, the values of the function become more negative and grow in magnitude. As $t$ approaches 0 from the negative side, the values become more positive and also grow in magnitude. The function does not approach a single value from both sides of 0, so the limit does not exist as $t \to 0$.

Conclusion:

$\lim_{{t \to 0}} \frac{6t - 1}{t} \text{ does not exist.}$

Would you like to explore this concept further, or would you like help with a different type of problem? Here are some related questions:

1. What happens if we evaluate the limit from one side only (left-hand or right-hand limit)?
2. How do you determine if a limit exists from a table of values?
3. Can limits exist if the function approaches infinity on one side?
4. What is the difference between an infinite limit and a limit that does not exist?
5. Can you use L'Hopital's Rule to evaluate this limit?

Tip: When you have a limit where the function behaves differently from either side of a point, check if there are any asymptotes or discontinuities at that point.