Chapter 2.2, Problem 30E

a.

To determine

To evaluate the given function for given values.

Answer to Problem 30E

  $\begin{array}{l}h(1)=0.557408\\ h(0.5)=0.37042\\ h(0.1)=0.334672\\ h(0.05)=0.333667\\ h(0.01)=0.333347\\ h(0.05)=0.333337\end{array}$

Explanation of Solution

Given:

The given function is $h(x)=\frac{\tan x-x}{x^3}$

Calculation:

Substituting the following values in the given function as follows-

  $\begin{array}{l}h(x)=\frac{\tan x-x}{x^3}\\ h(1)=0.557408\\ h(0.5)=0.37042\\ h(0.1)=0.334672\\ h(0.05)=0.333667\\ h(0.01)=0.333347\\ h(0.05)=0.333337\end{array}$

b.

To determine

To guess the value of given function.

Answer to Problem 30E

  $\frac{1}{3}$

Explanation of Solution

Given:

The given function is $\underset{x\to 0}{\lim }\frac{\tan x-x}{x^3}$

Calculation:

  $\begin{array}{l}\underset{x\to 0}{\lim }\frac{\tan x-x}{x^3}\\ \underset{x\to 0}{\lim }\frac{\tan x-x}{x^3}=0.33333=\frac{1}{3}\end{array}$

c.

To determine

To evaluate $h(x)$ for successively smaller values of $x$ and to explain whether your guess is correct.

Answer to Problem 30E

No.

Explanation of Solution

Given:

The given function is $\underset{x\to 0}{\lim }\frac{\tan x-x}{x^3}$

Calculation:

The following is the table of all smaller values-

  

The guess in part (b) does not seem to be correct.

d.

To determine

To graph the function $h$ in the viewing rectangle $[-1,1]$ by $[0,1]$ and continue to zoom until distortions are observed and then compare the results.

Explanation of Solution

Given:

The given function is $\underset{x\to 0}{\lim }\frac{\tan x-x}{x^3}$

Calculation:

The following are the graphs-

  

  

In part (C), it was observed that $\underset{x\to 0}{\lim }\frac{\tan x-x}{x^3}=0$

From the above graphs the limit does not approach to a specific value, hence the limit does not exist.