   Chapter 3.4, Problem 23E

Chapter
Section
Textbook Problem

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

To determine

To find:

limx(x2+ax-x2+bx)

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:limp(x)q(x)= limx p(x)limx q(x)

ii) Sum law: lim ∞[px+qx]=lim ∞p(x)+lim ∞q(x)

iii) Difference law: lim ∞[px-qx]=lim ∞p(x)-lim ∞q(x)

iv) Constant law: lim ∞c =c

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

3) Given:

limx(x2+ax-x2+bx)

4) Calculation:

Multiply numerator and denominator by (x2+ax+x2+bx),

limx(x2+ax-x2+bx)=limxx2+ax-x2+bxx2+ax+x2+bxx2+ax+x2+bx

=limxx2+ax-x2-bxx2+ax+x2+bx

=limx

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