   Chapter 3.4, Problem 41E

Chapter
Section
Textbook Problem

# Estimate the horizontal asymptote of the function f ( x ) = 3 x 3 + 500 x 2 x 3 + 500 x 2 + 100 x + 2000 Estimate the horizontal asymptote of the functionby graphing f for − 10 ≤ x ≤ 10 . Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?

To determine

To estimate:

a) Horizontal asymptote of the given function by graphing f for -10x10

b) Calculate the equation of the asymptote by evaluating the limit

c) Explain the discrepancy

Explanation

1) Concept:

Use the definition of horizontal asymptote

2) Definition:

Horizontal asymptote:

Horizontal asymptote: y=b

As x±, yb or limx±f(x)=b

3) Given:

fx=3x3+500x2x3+500x2+100x+2000

4) Calculation:

i. The graph of f for -10x10 is

From the graph it is observed that y=1 is a horizontal asymptote

ii. Consider the given function,

fx=3x3+500x2x3+500x2+100x+2000

Divide numerator and denominator by x3 and by using limit properties find the limit

=limx3x3+500x2x3+500x2+100x+2000

=limx3x3+500x2x3x3+500x2+100x+2000x3

Simplify,

=

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