Chapter 1.2, Problem 4QQ

a.

To determine

To calculate: The domain and the rage of the function $f\left(x\right)=\frac{\cos x}{x}$ .

Answer to Problem 4QQ

The domain is $\left(-\infty ,0\right)\cup \left(0,\infty \right)$ and the range is real numbers.

Explanation of Solution

Given Information:

The function is $f\left(x\right)=\frac{\cos x}{x}$ .

Calculation:

Consider the given function,

  $f\left(x\right)=\frac{\cos x}{x}$

To find the domain of the given equate the denominator to zero.

  $x=0$

So the domain is defined as,

  $\text{Domain}=\left(-\infty ,0\right)\cup \left(0,\infty \right)$

Now, find the range of the function.

The set of range is real numbers. There is no restrict value for range.

  $\text{range}=ℝ$

Therefore, the domain is $\left(-\infty ,0\right)\cup \left(0,\infty \right)$ and the range is real numbers.

b.

To determine

The given function $f\left(x\right)=\frac{\cos x}{x}$ is even function, odd function or neither.

Answer to Problem 4QQ

The given function is even function.

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{\cos x}{x}$ .

Formula Used:

A function is said to odd function if $f\left(-x\right)=f\left(x\right)$ , and the function is said to be even function if $f\left(-x\right)=-f\left(x\right)$ .

Calculation:

Consider the given equation,

  $f\left(x\right)=\frac{\cos x}{x}$

First check that the given function is even function.

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

So, the given function is not even function.

Check for the odd function.

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

So, the function is odd function.

Therefore, the given function is odd function.

c.

To determine

To calculate: The limit of the function as $\underset{x\to \infty }{\lim }f\left(x\right)\frac{\cos x}{x}$ .

Answer to Problem 4QQ

The limit of the function is zero.

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{\cos x}{x}$ .

Formula Used:

The cosine function is defined as,

i $-1\le \cos x\le 1$

Calculation:

Consider the given equation,

  $f\left(x\right)=\frac{\cos x}{x}$

Take the limit of the function.

  $\begin{array}{c}\underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\frac{\cos x}{x}\\ =\frac{\underset{x\to \infty }{\lim }\cos x}{\underset{x\to \infty }{\lim }x}\left[\cos x\le 1\right]\\ =\frac{1}{\infty }\\ =0\end{array}$

Therefore, the limit of the function is 0.

d.

To determine

The justification for the part (c) by using the squeeze theorem.

Answer to Problem 4QQ

The limit of the function is zero.

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{\cos x}{x}$ .

Calculation:

Consider the given equation,

The Squeeze theorem state that if $f\left(x\right)\le g\left(x\right)\le h\left(x\right)$ for all numbers at some point $x=k$ and $f\left(k\right)=h\left(k\right)$ , then $g\left(k\right)$ must be also equal to them.

Take the limit of the function.

  $\begin{array}{c}\underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\frac{\cos x}{x}\\ \le \underset{x\to \infty }{\lim }\frac{1}{x}\\ =0\end{array}$

Therefore, the limit of the previous part (a) has proved.