Chapter 1.2, Problem 4E

(a)

To determine

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

Answer to Problem 4E

The limit is $+\infty$

Explanation of Solution

Given information: The function is $f(x)=\frac{3x^3-x+1}{x+3}$

Graph:

The given function is:

  $f(x)=\frac{3x^3-x+1}{x+3}$

The graph of the function $f(x)=\frac{3x^3-x+1}{x+3}$ is shown below:

  

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

x	f(x)
1	0.75
10	230.08
100	29,125.25
1000	2,991,025.92
10000	299,910,025.99
100000	29,999,100,026.00

The function tends $+\infty$ as $x$ approaches infinity, which is consistent with what was seen on the graph.

With a graphing tool and a table, determine that the limit of $f(x)$ as $x$ approaches infinity equals $+\infty$ .

(b)

To determine

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

Answer to Problem 4E

The limit is $+\infty$

Explanation of Solution

Given information: The function is $f(x)=\frac{3x^3-x+1}{x+3}$

Graph:

The given function is:

  $f(x)=\frac{3x^3-x+1}{x+3}$

The graph of the function $f(x)=\frac{3x^3-x+1}{x+3}$ is shown below:

  

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

x	f(x)
-1	-0.50
-10	427.00
-100	30,926.79
-1000	3,009,026.08
-10000	300,090,026.01
-100000	30,000,900,026.00

The function tends $+\infty$ as $x$ approaches infinity, which is consistent with what was seen on the graph.

With a graphing tool and a table, determine that the limit of $f(x)$ as $x$ approaches infinity equals $+\infty$ .

(c)

To determine

To find: The horizontal asymptote

Answer to Problem 4E

There is no horizontal asymptote.

Explanation of Solution

Given information: The function is $f(x)=\frac{3x^3-x+1}{x+3}$

Calculation:

The given function is:

  $f(x)=\frac{3x^3-x+1}{x+3}$

From part a)

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

From part b)

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

Therefore, there are no asymptotes on the horizontal.