   Chapter 2.6, Problem 27E

Chapter
Section
Textbook Problem

Find the limit or show that it does not exist. lim x → ∞ ( 9 x 2 + x − 3 x )

To determine

To find: The value of limx(9x2+x3x).

Explanation

Limit Laws used: Suppose that c is a constant and the limits limxaf(x) and limxag(x)

exists, then

Limit law 1: limxa[f(x)+g(x)]=limxaf(x)+limxag(x)

Limit law 2: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 3: limxa[cf(x)]=climxaf(x)

Limit law 4: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 5: limxaf(x)g(x)=limxaf(x)limxag(x) if limxag(x)0

Limit law 6: limxa[f(x)]n=[limxaf(x)]n where n is a positive integer.

Limit law 7: limxac=c

Limit law 8: limxax=a

Limit law 9: limxaxn=an where n is a positive integer.

Limit law 10: limxaxn=an where n is a positive integer, if n is even, assume that a>0.

Limit law 11: limxaf(x)n=limxaf(x)n where n is a positive integer, if n is even, assume that limxaf(x)>0.

Theorem used: If r>0 is a rational number, then limx1xr=0.

Calculation:

Obtain the value of the function as x approaches infinity.

Consider the function f(x)=(9x2+x3x).

f(x)=(9x2+x3x)        =(9x2+x3x)1×(9x2+x+3x)(9x2+x+3x)        =((9x2+x)2(3x)2)(9x2+x+3x)        =9x2+x9x2(9x2+x+3x)

Simplify further f(x) by using elementary algebra,

f(x)=x(9x2+x+3x)=xx2(9+1x)+3x=xx2(9+1x)+3x=x|x|(9+1x)+3x[x2=|x|]

Since x>0, then |x|=x

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