Question

use a calculator in radian mode to complete the following table. (round to four decimal places as needed.)
\\( \

$$\begin{array} { c c } { \\theta } & { - 3 \\tan \\theta } \\\\ { 0.5 } & { - 1.6389 } \\\\ { 0.4 } & { - 1.2684 } \\\\ { 0.2 } & { - 0.6081 } \\\\ { 0.1 } & { - 0.3009 } \\end{array}$$

\\)
\\( f ( \theta ) = - \frac { 3 \tan \theta } { \theta } \\)
\\( \

$$\begin{array} { l } { 3.2778 } \\\\ { 3.1710 } \\\\ { 3.0405 } \\\\ { 3.0090 } \\end{array}$$

\\)
use a calculator in radian mode to complete the following table. what can be concluded about the value of \\( f ( \theta ) = - \frac { 3 \tan \theta } { \theta } \\) as \\( \theta \\) approaches 0?

Explanation:

Step1: Analyze the trend of \(f(\theta)\) values

As \(\theta\) gets closer to \(0\) (i.e., \(\theta = 0.5,0.4,0.2,0.1\) and approaching \(0\)), we observe the values of \(f(\theta)=-\frac{\theta}{3\tan\theta}\). The values of \(f(\theta)\) are \(3.2778,3.1710,3.0405,3.0090\) respectively.

Step2: Determine the limit behavior

As \(\theta\) approaches \(0\), from the pattern of the function values \(3.2778,3.1710,3.0405,3.0090\) (getting closer and closer to \(3\)).

Answer:

As \(\theta\) approaches \(0\), \(f(\theta)=-\frac{\theta}{3\tan\theta}\) approaches \(3\).