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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

35-40 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 and check the work by graphing the curve and estimating asymptotes.

Explanation

1) Concept:

Use definition of horizontal and vertical asymptote.

2) Definitions:

Horizontal asymptote:

Horizontal asymptote: y=b

As x±,yb or limx±f(x)=b

Vertical asymptote:

Vertical asymptote: x=c

As xc,y± or limxcf(x)=±

3) Given:

y=1+x4x2-x4

4) Calculation:

Consider the given function,

y=1+x4x2-x4

Divide numerator and denominator by x4 and by using limit properties find the limit

limx1+x4x2-x4

limx1+x4x4x2-x4x4

Simplify,

limx1x4+11x2-1

Apply limit separately,

limx1x4+limx1limx1x2-limx1

Since, 1x4, 1x20 as x

=0+10-1

=1-1

=-1

As limit x±  is same for even functions, therefore

limx ±1+x4x2-x4=-1

Therefore, the line y=-1 is horizontal asymptote

The vertical asymptote is likely to occur when denominator x2-x4=0

Solving for x

x21-x2=0

x21-x1+x=0

x=0, 1, -1

At x=0, 1, -1

limx0-1+x4<

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