Let's break this down and complete the table step by step for the function $f(x) = \frac{3 \sin(x)}{x}$.

Steps:

1. Understand the Limit Expression: $\lim_{x \to 0} \frac{3 \sin(x)}{x}$ This limit is very similar to the standard limit: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ By multiplying by 3, we expect the limit of $\frac{3 \sin(x)}{x}$ as $x \to 0$ to be 3.

2. Values of $f(x)$ at Specific Points: Now, we can compute $f(x)$ at the given $x$ values. For each $x$, use the formula: $f(x) = \frac{3 \sin(x)}{x}$

   • For $x = -0.1$, we know $f(x) = 2.99500$ (already provided).
   • For $x = -0.01$, the correct value was not given, so we'll recompute it.
   • For $x = -0.001$, recompute the correct value.
   • For $x = 0$, we know from the limit calculation that $f(0) = 3$ (since the limit approaches 3).
   • For $x = 0.001$, recompute the correct value.
   • For $x = 0.01$, recompute the correct value.
   • For $x = 0.1$, we know $f(x) = 2.99500$ (already provided).

3. Compute the Missing Values:

I'll calculate these values for you now.Here are the calculated values for $f(x) = \frac{3 \sin(x)}{x}$ at the given $x$-values:

$x$	$f(x)$
-0.1	2.99500
-0.01	2.99995
-0.001	2.99999
0 (limit)	3.00000 (as $x \to 0$)
0.001	2.99999
0.01	2.99995
0.1	2.99500

Conclusion:

The table is now complete, and the limit as $x \to 0$ for $\frac{3 \sin(x)}{x}$ is $3.00000$.

Tip:

Whenever dealing with limits involving $\sin(x)$ as $x \to 0$, it's helpful to remember that $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$.

Would you like more details or clarifications on this? Here are 5 related questions to explore:

1. What is the limit of $\frac{\sin(x)}{x}$ when $x$ approaches infinity?
2. How do we derive the standard limit $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$?
3. What happens to the function $\frac{3 \sin(x)}{x}$ for very large values of $x$?
4. Can L'Hopital's rule be applied to this problem? How?
5. How would the table look if the function was $\frac{2 \sin(x)}{x}$ instead of $3 \sin(x)$?

Tip: For small angles, $\sin(x) \approx x$, which simplifies the limit calculations.