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 𝜃→0 sin(5𝜃)tan(2𝜃)

To estimate the value of the limit $\lim_{\theta \to 0} \sin(5\theta) \tan(2\theta)$ using a table of values, follow these steps:

1. Create a Table of Values: Choose values of $\theta$ that are close to 0, both positive and negative. For example, you might choose $\theta = -0.1, -0.05, -0.01, 0, 0.01, 0.05, 0.1$.

2. Calculate $\sin(5\theta)$ and $\tan(2\theta)$ for Each $\theta$: Use a calculator to find the values of $\sin(5\theta)$ and $\tan(2\theta)$ for each chosen $\theta$.

3. Multiply the Results: For each $\theta$, multiply the values of $\sin(5\theta)$ and $\tan(2\theta)$ to get $\sin(5\theta) \tan(2\theta)$.

4. Observe the Trend: As $\theta$ approaches 0, observe the trend in the values of $\sin(5\theta) \tan(2\theta)$.

Here is an example table:

$\begin{array}{|c|c|c|c|} \hline \theta & \sin(5\theta) & \tan(2\theta) & \sin(5\theta) \tan(2\theta) \\ \hline -0.1 & -0.4794 & -0.2027 & 0.0971 \\ -0.05 & -0.2474 & -0.1003 & 0.0248 \\ -0.01 & -0.04998 & -0.02000 & 0.0010 \\ 0 & 0 & 0 & 0 \\ 0.01 & 0.04998 & 0.02000 & 0.0010 \\ 0.05 & 0.2474 & 0.1003 & 0.0248 \\ 0.1 & 0.4794 & 0.2027 & 0.0971 \\ \hline \end{array}$

From the table, as $\theta$ approaches 0, the value of $\sin(5\theta) \tan(2\theta)$ approaches 0.

5. Confirm Graphically: If you have a graphing device, plot the function $y = \sin(5\theta) \tan(2\theta)$ and observe the behavior as $\theta$ approaches 0. The graph should show that the function approaches 0 as $\theta$ approaches 0.

Therefore, the estimated value of the limit is:

$\lim_{\theta \to 0} \sin(5\theta) \tan(2\theta) = 0.000$