Chapter 1.2, Problem 3E

(a)

To determine

To find: The limit of $\underset{x\to \infty }{\lim }f(x)$

Answer to Problem 3E

The limit is 0

Explanation of Solution

Given information: The function is $f(x)=\frac{e^{-x}}{x}$

Graph:

The given function is:

  $f(x)=\frac{e^{-x}}{x}$

The graph of the function $f(x)=\frac{e^{-x}}{x}$ is shown below:

  

The table of values and $f(x)$ approaches that limit:

x	f(x)
1	0.3678794
2	0.0676676
4	0.0045789
8	0.0000419
16	0.0000000
32	0.0000000

  $f$ can be written as:

  $\begin{array}{c}f(x)=\frac{e^{-x}}{x}\\ =\frac{1}{x{e}^x}\end{array}$

A fraction in which the numerator is constant and the denominator increases without bounds ,Indeed the quotient must tend to 0. Therefore, the function $f$ tends to 0 as $x\to +\infty$

With a graphing tool and a table, determine that the limit of $e^{-x}$ as $x$ approaches infinity equals 0.

(b)

To determine

To find: The limit of $\underset{x\to -\infty }{\lim }f(x)$

Answer to Problem 3E

The limit is 0.

Explanation of Solution

Given information: The function is $f(x)=\frac{e^{-x}}{x}$

Graph:

The given function is:

  $f(x)=\frac{e^{-x}}{x}$

The graph of the function $f(x)=\frac{e^{-x}}{x}$ is shown below:

  

The table of values and $f(x)$ approaches that limit:

x	f(x)
-10	-2202.65
-11	-5443.1
-12	-13,562.9
-13	-34,031.8
-14	-85,900.3
-15	-217,934
-16	-555,382

With a graphing tool and a table, determine that the limit of $e^{-x}$ as $x$ approaches infinity equals 0.

(c)

To determine

To find: The horizontal asymptote

Answer to Problem 3E

  $\underset{x\to \infty }{\lim }f(x)=0$

  $y=0$ is the horizontal asymptote.

And,

  $\underset{x\to -\infty }{\lim }f(x)=-\infty$

There isn't a horizontal asymptote in the opposite direction.

Explanation of Solution

Given information: The function is $f(x)=\frac{e^{-x}}{x}$

Calculation:

The given fraction is:

  $f(x)=\frac{e^{-x}}{x}$

From part a)

  $\underset{x\to \infty }{\lim }f(x)=0$

  $y=0$ is the horizontal asymptote.

From part b)

  $\underset{x\to -\infty }{\lim }f(x)=-\infty$ (which is not constant)

There isn't a horizontal asymptote in the opposite direction.