   Chapter 2.6, Problem 47E

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 = 5 + 4 x x + 3

To determine

To find: The horizontal and vertical asymptotes of y=5+4xx+3.

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)

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=5+4xx+3 as follows

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)=5+4xx+3.

Divide both the numerator and the denominator by the highest power of x in the denominator. That is, x0.

f(x)=5+4xxx+3x=5x+4xxxx+3x=5x+41+3x

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

limx5+4xx+3limx5x+41+3x                    =limx(5x+4)limx(1+3x)[by limit law 5]

Apply the appropriate laws and simplify further.

limx5+4xx+3=limx(5x)+limx(4)limx(1)+limx(3x)[by limit law 1]=5limx(1x)+41+3limx(1x)[by limit law 3,7]

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

limx5+4xx+3=5(0)+41+3(0)=41=4

Therefore, the function f(x)=5+4xx+3 is approaching 4 as x approaches infinity

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