Chapter 5.3, Problem 87E

a.

To determine

Find the domain of the function.

Answer to Problem 87E

  $\boxed{R-\left\{0\right\}}$

Explanation of Solution

Given information:

Consider the function $f\left(x\right)=\frac{\sin x}{x}$ and its graph, shown in the figure below,

  

Calculation:

Consider the function,and graph,

  $f\left(x\right)=\frac{\sin x}{x}$

  

  $\begin{array}{l}put,x=0\\ y=\frac{\sin x}{x}\\ y=\frac{\sin 0}{0}=\frac{0}{0}=0\end{array}$

This is not defined , function $f$ is defined everywhere except $x=0$

Hence, the domain of the function is, $\boxed{R-\left\{0\right\}}$

b.

To determine

Identify any symmetry and any asymptotes of the graph.

Answer to Problem 87E

  $\boxed{y-axis}$

Explanation of Solution

Given information:

Consider the function $f\left(x\right)=\frac{\sin x}{x}$ and its graph, shown in the figure below,

  

Calculation:

Consider the function,and graph,

  $f\left(x\right)=\frac{\sin x}{x}$

The graph has no asymptotes because it does not approach to a particular value,

For symmetry replace $x$ to $-x$

  $\begin{array}{l}f\left(-x\right)=\frac{\sin \left(-x\right)}{-x}\\ =\frac{-\sin x}{-x}\\ =\frac{\sin x}{x}\end{array}$

Which is equal to $f\left(x\right)$ means the function is even and symmetrical about $y-axis$ .

Hence the function is symmetrical about $\boxed{y-axis}$

c.

To determine

Describe the behaviour of the function as $x\to 0$ .

Answer to Problem 87E

  $\boxed{f\left(x\right)=\frac{\sin x}{x}\to 1}$

Explanation of Solution

Given information:

Consider the function $f\left(x\right)=\frac{\sin x}{x}$ and its graph, shown in the figure below,

  

Calculation:

Consider the function,and graph,

  $f\left(x\right)=\frac{\sin x}{x}$

  

According to the graph $x\to 0$ then function $f\left(x\right)=\frac{\sin x}{x}\to 1$

Hence, the behaviour of the function as $x\to 0$ is $\boxed{f\left(x\right)=\frac{\sin x}{x}\to 1}$

d.

To determine

Find the number of solutions.

Answer to Problem 87E

  $\boxed{4}$ , $\boxed{-\pi ,-2\pi ,\pi ,2\pi }$

Explanation of Solution

Given information:

Consider the function $f\left(x\right)=\frac{\sin x}{x}$ and its graph, shown in the figure below,

  

How many solutions does the equation $\frac{\sin x}{x}=0$ have in the interval $\left[-8,8\right]$ ? Find the solutions.

Calculation:

Consider the function,and graph,

  $f\left(x\right)=\frac{\sin x}{x}$

  

The function $f\left(x\right)=\frac{\sin x}{x}$ is equal to zero only when $\sin x=0$ and $x\ne 0$

So $x=0$ is not the not the solution of function,

Now with maple the graph of $f\left(x\right)=\frac{\sin x}{x}$ in the interval $\left[-8,8\right]$ will be,

  

The function cuts the $x-axis$

  $4$ times,

Hence, the solutions are $\boxed{4}$ which are $\boxed{-\pi ,-2\pi ,\pi ,2\pi }$