   Chapter 2.6, Problem 61E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find the limits as x → ∞ and as x → –∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.Example 12 FIGURE 13y = (x – 2)4 (x + 1)3 (x – 1) y = x4 – x6

To determine

To find: The limit of the function y=x4x6 as x approaches infinity and negative infinity and obtain the x and y intercepts and sketch the rough graph of the function f(x)=x4x6.

Explanation

Result used:

Limit Laws: Suppose that c is a constant and the limits limxaf(x) and limxag(x) exists, then

Limit law 1: limxa[f(x)+g(x)]=limxaf(x)+limxag(x)

Limit law 2: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 3: limxa[cf(x)]=climxaf(x)

Theorem used:

1. If r>0 is a rational number, then limx1xr=0.

2. If r>0 is a rational number such that xr is defined, then limx1xr=0.

Note:

The limit law limxaf(x)g(x)=limxaf(x)limxag(x) is valid for one sided limits. That is, for a=±, and also for infinite limits using the rules b×= if b>0.

Calculation:

Obtain the x and y intercepts.

Consider the function, f(x)=x4x6.

The y-intercept is computed as follows,

Plug x=0 in the function f(x),

f(0)=(0)4(0)6=0

Thus, the y-intercept is 0.

The x-intercept is computed as follows,

Plug f(x)=0 in the function f(x)=x4x6,

x4x6=0x4(1x2)=0x4(1x)(1+x)=0x=0,1 and 1

Thus, the x intercepts are 0, 1 and −1.

Compute the value of the function as x approaches infinity.

Take the limit of f(x) as x approaches infinity.

limx(x4x6)=limxx6(1x21)

Here, x6 goes to infinity and 1x21 goes to −1 as x approaches infinity. Thus, the product f(x)=x6(1x21) approaches negative infinity as x approaches infinity. That is,

limx(x4x6)=limxx6limx(1x21)[by note]=limxx6[limx(1x2)limx(1)][by limit law 2]=limxx6[(0)limx(1)][by theorem]=()

=

Thus, the function f(x)=x6(1x21) approaches negative infinity as x approaches infinity

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