   Chapter 10.5, Problem 3E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In Problems 1-4, a function and its graph are given. Use the graph to find each of the following, if they exist. Then confirm your results analytically.(a) vertical asymptotes(b) lim x → ∞ f ( x ) (c) lim x → − ∞ f ( x ) (d) horizontal asymptotes f ( x ) = 3 ( x 4 + 2 x 3 + 6 x 2 + 2 x + 5 ) ( x 2 − 4 ) 2 (a)

To determine

The vertical asymptote for the function f(x)=3(x4+2x3+6x2+2x+5)(x24)2 from the provided graph, Also, verify the result analytically.

Explanation

Given Information:

The provided graph of the function f(x)=3(x4+2x3+6x2+2x+5)(x24)2 is,

.

Explanation:

A vertical asymptote of a function f(x) is a line x=a such that f(a)=.

Consider the provided graph,

From the graph, at x=2 and x=2 the graph of the function decreases and increases indefinitely.

Thus, the vertical asymptotes are the lines x=2 and x=2.

A vertical asymptote of a rational function h(x)=f(x)g(x) is x=a where g(a)=0 and f(a)0.

Now, consider the function f(x)=3(x4+2x3+6x2+2x+5)(x24)2.

Set the denominator of the function equal to 0.

(x24)2=0x24=0x24+4=0+4x2=4

Take square root on both sides.

x=±4=±2

Thus, the possible vertical asymptotes are the lines x=2 and x=2

(b)

To determine

The value of limxf(x) for the function f(x)=3(x4+2x3+6x2+2x+5)(x24)2 from the provided graph, Also, verify the result analytically.

(c)

To determine

The value of limxf(x) for the function f(x)=3(x4+2x3+6x2+2x+5)(x24)2 from the provided graph, Also, verify the result analytically.

(d)

To determine

The horizontal asymptote for the function f(x)=3(x4+2x3+6x2+2x+5)(x24)2 from the provided graph, Also, verify the result analytically.

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