Question

consider the following. \\(\lim\limits_{x \to 0} \frac{\sin(2x)}{x}\\) create a table of values for the function. (round your answers to four decimal places.) use the table to estimate the limit. use a graphing utility to graph the function to confirm your result. (round your answer to four decimal places.) \\(\lim\limits_{x \to 0} \frac{\sin(2x)}{x} \approx \underline{\quad\quad}\\)

Explanation:

Step1: Define the function

$f(x) = \frac{\sin(2x)}{x}$

Step2: Calculate $f(-0.1)$

$\frac{\sin(2\times(-0.1))}{-0.1} = \frac{\sin(-0.2)}{-0.1} \approx 1.9867$

Step3: Calculate $f(-0.01)$

$\frac{\sin(2\times(-0.01))}{-0.01} = \frac{\sin(-0.02)}{-0.01} \approx 1.9999$

Step4: Calculate $f(-0.001)$

$\frac{\sin(2\times(-0.001))}{-0.001} = \frac{\sin(-0.002)}{-0.001} \approx 2.0000$

Step5: Calculate $f(0.001)$

$\frac{\sin(2\times0.001)}{0.001} = \frac{\sin(0.002)}{0.001} \approx 2.0000$

Step6: Calculate $f(0.01)$

$\frac{\sin(2\times0.01)}{0.01} = \frac{\sin(0.02)}{0.01} \approx 1.9999$

Step7: Calculate $f(0.1)$

$\frac{\sin(2\times0.1)}{0.1} = \frac{\sin(0.2)}{0.1} \approx 1.9867$

Step8: Estimate the limit

As $x$ approaches 0, $f(x)$ approaches 2.0000.

Answer:

Completed Table:

$x$	$f(x)$
$-0.01$	1.9999
$-0.001$	2.0000
$0.001$	2.0000
$0.01$	1.9999
$0.1$	1.9867

$\lim_{x \to 0} \frac{\sin(2x)}{x} \approx 2.0000$