Chapter 4.6, Problem 75E

a.

To determine

To complete: The table and round the results to four decimal places.

Answer to Problem 75E

The values are filled in the tables below.

Explanation of Solution

Given Information: Function is $f(x)=\frac{\tan x}{x}$ .

Calculation:

Calculating the values and arranging in tables,

  $\begin{array}{lllll}x\hfill & -1\hfill & -0.1\hfill & -0.01\hfill & -0.001\hfill \\ \frac{\tan x}{x}\hfill & 1.5574\hfill & 1.0033\hfill & 1.0\hfill & 1.0\hfill \end{array}$

  $\begin{array}{llllll}x\hfill & 0\hfill & 0.001\hfill & 0.01\hfill & 0.1\hfill & 1\hfill \\ \frac{\tan x}{x}\hfill & Undefined\hfill & 1.0\hfill & 1.0\hfill & 1.0033\hfill & 1.5574\hfill \end{array}$

Thus, the values are filled in the tables.

b.

To determine

To use: The graphing utility to graph $f$ and then to describe the behavior of graph as $x$ approaches $0$ using zoom and trace feature.

Answer to Problem 75E

As $x\to 0,f(x)\to 1$

.

Explanation of Solution

Given Information: Function is $f(x)=\frac{\tan x}{x}$ .

Graph:

  

Interpretation:

It can be clearly observed in the graph that as $x\to 0,f(x)\to 1$

.

Thus, as $x\to 0,f(x)\to 1$

.

c.

To determine

To write: A brief statement regarding the value of the ratio based on the results in previous parts.

Answer to Problem 75E

As $x$ approaches $0$ the value of ratio approaches $1$ .

Explanation of Solution

Given Information: Function is $f(x)=\frac{\tan x}{x}$ .

Calculation:

Combining the results from previous parts, tables and graph signify the same fact that as $x$ approaches $0$ the value of ratio approaches $1$ .

Thus, as $x$ approaches $0$ the value of ratio approaches $1$ .