   Chapter 3.R, Problem 11E

Chapter
Section
Textbook Problem

# 7-12 Find the limit. lim x →   ∞ ( 4 x 2 + 3 x − 2 x )

To determine

To find:

The limit limx4x2+3x-2x

Explanation

1) Concept:

To evaluate the limit at infinity of any rational function, divide 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. Constant law: lim ∞c =c

iv. Root law:limxp(x)n=limxp(x)n

3) Theorem:

If r > 0 is a rational number such that xr is defined for all x, then limx 1xr=0

4) Given:

limx4x2+3x-2x

5) Calculation:

Multiply numerator and denominator by the conjugate of radical that is 4x2+3x+2x

Simplify the given equation,

limx4x2+3x-2x=limx 4x2+3x-2x4x2+3x+2x4x2+3x+2x

Simplifying the numerator by using the difference of square formula

a2-b2=a+b(a-b)

=limx4x2+3x2-2x24x2+3x+2x

= limx3x4x2+3x+2x

Here, highest power of denominator is x2=x, so divide numerator and denominator by x2

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