   Chapter 3.4, Problem 15E

Chapter
Section
Textbook Problem

# 9-32 Find the limit or show that it does not exist. lim x → ∞ ( 2 x 2 + 1 ) 2 ( x − 1 ) 2 ( x 2 + x )

To determine

To find:

limx2x2+12x-12(x2+x)

Explanation

1) Concept:

To evaluate the limit at infinity of any rational function, first divide both the numerator and denominator by the highest power of x that occurs in the denominator.

2) Formula:

i) Quotient law: limpxqx=limp(x)limq(x)

ii) Sum law: lim[px+qx]=limp(x)+limq(x)

iii) Difference law: lim[px-qx]=limp(x)-limq(x)

iv) Constant multiple law: lim ∞c p(x)=clim ∞p(x)

v) Constant law: lim ∞c =c

vi) Product law: Sum law: lim[px.qx]=limp(x).limq(x)

vii) Power law: limpxn=limx pxnwhere n is integer.

2) Given:

limx2x2+12x-12(x2+x)

3) Calculation:

Here, highest power of denominator is x4, divide numerator and denominator by x4

limx2x2+12x-12x2+x=limx2x2+12x4x-12x2+xx4

=limx2x2

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