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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

48-51 Find the horizontal asymptotes of the curve and use them, together with concavity and intervals of increase and decrease, to sketch the curve.

y = 1 x 1 + x

To determine

To find:

i. The horizontal asymptotes of the curve

ii. Sketch the curve using horizontal asymptote, concavity and intervals of increase and decrease

Explanation

1) Concept:

Use the definition of horizontal asymptote, find intervals of increase and decrease and concavity to sketch the curve

2) Definition:

Horizontal asymptote:

Horizontal asymptote: y=b

As x±,yb or limx±f(x)=b

3) Given:

y=1-x1+x

4) Calculation:

i. Consider the given function,

y=1-x1+x

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

=limx1-x1+x

=limx1-xx1+xx

Simplify,

=limx1x-11x+1

Apply limit separately,

=limx1x-limx1limx1x+limx1

Since, 1x0 as x

Therefore,

=0-10+1

=-11

=-1

In computing the limit as x- for x<0 we have,

limx-1+x1-x=-1

Therefore, the line y=-1 is horizontal asymptote

The vertical asymptote is likely to occur when denominator 1+x=0x=-1

Now,

limx-1+1-x1+x=+

limx-1-1-x1+x=-

Therefore, x=-1 is vertical asymptote

ii

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