   Chapter 2.6, Problem 50E

Chapter
Section
Textbook Problem

Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check: your work by graphing the curve and estimating the asymptotes. y = 1 + x 4 x 2 − x 4

To determine

To find: The horizontal and vertical asymptotes of y=1+x4x2x4.

Explanation

Definition used:

Limit Laws used: 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)

Limit law 4:limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 5:limxaf(x)g(x)=limxaf(x)limxag(x) if limxag(x)0

Limit law 6:limxa[f(x)]n=[limxaf(x)]n where n is a positive integer.

Limit law 7:limxac=c

Limit law 8: limxax=a

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

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

Calculation:

Horizontal Asymptote:

Obtain the horizontal asymptote of the function.y=1+x4x2x4.

Recall the definition of horizontal asymptote, “the line y=L is called a horizontal asymptote of the curve y=f(x) if either limxf(x)=L and limxf(x)=L”.

Consider the function f(x)=1+x4x2x4.

Divide both the numerator and the denominator by the highest power of x in the denominator. That is, x40.

f(x)=1+x4x4x2x4x4=1x4+x4x4x2x4x4x4=1x4+11x21

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

limx1+x4x2x4limx1x4+11x21                    =limx(1x4+1)limx(1x21)[by limit law 5]

Apply the appropriate laws and simplify further.

limx1+x4x2x4=limx(1x4)+limx(1)limx(1x2)limx(1)[by limit law 1,2]=limx(1x4)+1limx(1x2)1[by limit law 3,7]

Since r>0, apply the theorem1 stated above to compute the value of the limit function.

limx1+x2x2x4=(0)+1(0)1=11=1

Therefore, the function f(x)=1+x4x2x4 is approach 1 as x approaches infinity. That is, limx1+x4x2x4=1.

Since, limx1+x4x2x4=1, the line y=1 is called a horizontal asymptote of the curve

y=1+x4x2x4.

Thus, the horizontal asymptote is 1.

Compute the limit of f(x) as x approaches negative infinity.

limx1+x4x2x4limx1x4+11x21                    =limx(1x4+1)limx(1x21)[by limit law 5]

Apply the appropriate laws and simplify further

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