Chapter 5.3, Problem 88E

a.

To determine

Find the domain of the function.

Answer to Problem 88E

  $\boxed{x\in R-\left\{0\right\}}$

Explanation of Solution

Given information:

Consider the function $f\left(x\right)=\cos \frac{1}{x}$ and its graph, shown in the figure below,

  

Calculation:

Consider the function,and graph,

  $f\left(x\right)=\cos \frac{1}{x}$

The domain of the function is given by,

  $x\in R-\left\{0\right\}at,x=0$

  $f\left(0\right)=\cos \frac{1}{0}$

Undefined,

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

b.

To determine

Identify any symmetry and any asymptotes of the graph.

Answer to Problem 88E

  $\boxed{y-axis}$ , $\boxed{x=0,y=1}$

Explanation of Solution

Given information:

Consider the function $f\left(x\right)=\cos \frac{1}{x}$ and its graph, shown in the figure below,

  

Calculation:

Consider the function,and graph,

  $f\left(x\right)=\cos \frac{1}{x}$

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

For symmetry replace $x$ to $-x$

  $\begin{array}{l}f\left(x\right)=\cos \frac{1}{x}\\ =f\left(-x\right),at,x=0\\ =f\left(0\right)=\cos \frac{1}{0}\\ undefined,atx=\pm \infty \\ \underset{x\to \pm \infty }{\lim }\cos \frac{1}{x}\to 1\\ \boxed{x=0,y=1}\end{array}$

The graph has asymptotes $\boxed{x=0,y=1}$

symmetrical about $y-axis$ .

Hence the function is symmetrical about and has asymptotes $\boxed{y-axis}$ , $\boxed{x=0,y=1}$

c.

To determine

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

Answer to Problem 88E

  $\boxed{f\left(x\right)=\left[-1,1\right]}$

Explanation of Solution

Given information:

Consider the function $f\left(x\right)=\cos \frac{1}{x}$ and its graph, shown in the figure below,

  

Calculation:

Consider the function,and graph,

  $f\left(x\right)=\cos \frac{1}{x}$

  

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

  $\begin{array}{l}=\underset{x\to \pm \infty }{\lim }\cos \frac{1}{x}\\ =\cos \left(\pm \infty \right)\end{array}$

Hence,the function reciprocate between the limits, $\boxed{f\left(x\right)=\left[-1,1\right]}$

d.

To determine

Find the number of solutions.

Answer to Problem 88E

  $\boxed{x=\frac{2}{\pi },\frac{2}{3\pi }\mathrm{....}\frac{2}{\left(2n-1\right)\pi }}$

Explanation of Solution

Given information:

Consider the function $f\left(x\right)=\cos \frac{1}{x}$ and its graph, shown in the figure below,

  

How many solutions does the equation $\cos \frac{1}{x}=0$ have in the interval $\left[-1,1\right]$ ? Find the solutions.

Calculation:

Consider the function,and graph,

  $f\left(x\right)=\cos \frac{1}{x}$

  $\begin{array}{l}\cos \frac{1}{x}=0\\ \frac{1}{x}=\frac{\pi }{2},\frac{3\pi }{2}\\ x=\frac{2}{\pi },\frac{2}{3\pi }\mathrm{....}\frac{2}{\left(2n-1\right)\pi }\end{array}$

Hence infinite number of solution exists, which are $\boxed{x=\frac{2}{\pi },\frac{2}{3\pi }\mathrm{....}\frac{2}{\left(2n-1\right)\pi }}$ .

e.

To determine

Find the equation have a greatest solution.

Answer to Problem 88E

  $\boxed{x\approx 0.6366}$

Explanation of Solution

Given information:

Consider the function $f\left(x\right)=\cos \frac{1}{x}$ and its graph, shown in the figure below,

  

Does the equation $\cos \frac{1}{x}=0$ have a greatest solution? If so, then approximate the solution. If not, then explain why.

Calculation:

Consider the function,and graph,

  $f\left(x\right)=\cos \frac{1}{x}$

  

The greatest solution is appears at $x\approx 0.6366$

  $\boxed{x\approx 0.6366}$