Chapter 13, Problem 1T

(a)

To determine

To find: The value of $\underset{x\to 0}{\lim }\frac{x}{\sin 2x}$ by using of table of values.

Answer to Problem 1T

The value of $\underset{x\to 0}{\lim }\frac{x}{\sin 2x}$ is 0.5.

Explanation of Solution

Given:

The limit is $\underset{x\to 0}{\lim }\frac{x}{\sin 2x}$.

Calculation:

The given function is,

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

Substitute different values of x which are smaller than 0 in above function $f\left(x\right)$ and form the table,

x	$f\left(x\right)$
$-0.1$	0.50334895
$-0.01$	0.50003333
$-0.001$	0.50000033
$-0.0001$	0.50000003
$-0.00001$	0.5

Substitute different values of x which are greater than 0 on above function $f\left(x\right)$ and form the table,

x	$f\left(x\right)$
$0.1$	0.50334895
$0.01$	0.50003333
$0.001$	0.50000033
$0.0001$	0.50000003
$0.00001$	0.5

From both the above tables, it is observed that at the value of $x=\pm 0.0001$, the value of function $f\left(x\right)$ is 0.5.

Thus, the value of limit $\underset{x\to 0}{\lim }\frac{x}{\sin 2x}$ is 0.5.

(b)

To determine

To sketch: The graph of $\underset{x\to 0}{\lim }\frac{x}{\sin 2x}$ by graphing calculator to prove that the value of $\underset{x\to 0}{\lim }\frac{x}{\sin 2x}$ is 0.5.

Explanation of Solution

Sketch the graph of function $f\left(x\right)=\frac{x}{\sin 2x}$ by graphing calculator to check the limit,

Figure (1)

From the above Figure (1), it is proved that the value of $\underset{x\to 0}{\lim }\frac{x}{\sin 2x}$ is 0.5.