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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 = x x 2 + 1

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=xx2+1

4) Calculation:

i. Consider the given function,

y=xx2+1

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

=limxxx2+1

=limxxxx2x2+1x2

Simplify,

=limx11+1x2

Apply limit separately,

=limx1limx1+limx1x2

Since, 1x20 as x

Therefore,

=11+0

=11

=1

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

limx--xx2+1=-1

Therefore, the line y=±1 are horizontal asymptotes

The vertical asymptote is likely to occur when denominator x2+1=0

But there is no such x for which denominator is 0

So, it has no vertical asymptote

ii

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