Chapter 11, Problem 35CR

(a)

To determine

To find: The limit (if it exists) graphically by using graphing utility to graph the function.

Answer to Problem 35CR

  $\underset{x\to 0}{\lim }{e}^{-2/x}=0$

Explanation of Solution

Given information: $\underset{x\to 0}{\lim }{e}^{-2/x}$

Graph:

  

Interpretation:

From the graph, as x gets closer and closer to 0, $f(x)=e^{-2/x}$ gets closer and closer to 0 from the left and from the right.

Thus, $\underset{x\to 0}{\lim }{e}^{-2/x}=0$ .

(b)

To determine

To find: The limit (if it exists) numerically by using the table feature of the graphing utility to create a table.

Answer to Problem 35CR

  $\underset{x\to 0}{\lim }{e}^{-2/x}=0$

Explanation of Solution

Given information: $\underset{x\to 0}{\lim }{e}^{-2/x}$ .

Formula used: Construct a table that shows $f(x)=e^{-2/x}$ for two sets of x -values − one set that approaches 0 from the left and one that approaches 0 from the right.

Calculation:

x	$-0.1$	$-0.01$	$-0.2$	0	0.001	0.01	0.1
$f(x)=e^{-2/x}$	485165195.4	$\begin{array}{l}7.225937768\\ \times {10}^{86}\end{array}$	22026.46579	?	0	$\begin{array}{l}1.383896527\\ \times {10}^{-87}\end{array}$	$\begin{array}{l}2.061153622\\ \times {10}^{-9}\end{array}$

From the table,

As x approaches 0 from the left, $f(x)=e^{-2/x}$ approaches 0.

As x approaches 0 from the right, $f(x)=e^{-2/x}$ approaches 0.

So, it appears that the closer x gets to 0, the closer $f(x)=e^{-2/x}$ gets to 0.

Thus, $\underset{x\to 0}{\lim }{e}^{-2/x}=0$ .