   Chapter 3.4, Problem 51E

Chapter
Section
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 x2 and by using limit properties find the limit

=limxxx2+1

=limxxx2x2x2+1x2

Simplify,

=limx1x1+1x2

Apply limit separately,

=limx1xlimx1+limx1x2

Since, 1x, 1x20 as x

Therefore,

=01+0

=0

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

limx--xx2+1=0

Therefore, the line y=0 is horizontal asymptotes

The vertical asymptote is likely to occur when the denominator is 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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