Chapter 12.4, Problem 71E

To determine

Complete the table below to verify that $\underset{x\to \infty }{\lim }\left(1/x\right)=0$ .

Make a conjecture about $\underset{x\to \infty }{\lim }\left(1/x\right)$ .

Answer to Problem 71E

  $\begin{array}{ccccccc}x& 10°& {10}^1& {10}^2& {10}^3& {10}^4& {10}^5\\ \frac{1}{x}& 1& \frac{1}{10}& \frac{1}{100}& \frac{1}{1000}& \frac{1}{10,000}& \frac{1}{100,000}\end{array}$

The limit $\underset{x\to \infty }{\lim }\frac{1}{x}$ does not exist.

Explanation of Solution

Given information:

Use a graphing utility to complete the table below to verify that $\underset{x\to \infty }{\lim }\left(1/x\right)=0$ .

  $\begin{array}{ccccccc}x& 10°& {10}^1& {10}^2& {10}^3& {10}^4& {10}^5\\ \frac{1}{x}& & & & & & \end{array}$

Make a conjecture about $\underset{x\to \infty }{\lim }\left(1/x\right)$ .

Calculation:

First, construct a table of values using $TI$

  $83$ plus graphing calculator.

Follow the step by step procedure:

Input the function $f(x)=\frac{1}{x}$ using the key $\boxed{Y=}$ .

  

To display the values of the function in a table, open the TABLE SETUP menu by pressing $\boxed{2nd}$ and set Independent to Ask by placing the cursor over it and pressing $\boxed{ENTER}$ .

  

Press $\boxed{2nd}$ and input the values of the independent variable manually. Press $\boxed{ENTER}$ between to input values.

  

We see that the value of $Y_1$ get closer to $0$ as $X$ increases.

  $\begin{array}{ccccccc}x& 10°& {10}^1& {10}^2& {10}^3& {10}^4& {10}^5\\ \frac{1}{x}& 1& \frac{1}{10}& \frac{1}{100}& \frac{1}{1000}& \frac{1}{10,000}& \frac{1}{100,000}\end{array}$

So, $\underset{x\to \infty }{\lim }\frac{1}{x}=0$

The value of $\frac{1}{x}$ is incresaing without bound as $X$ approaches to $0$ .

Hence, the limit $\underset{x\to \infty }{\lim }\frac{1}{x}$ does not exist.