   Chapter 2.6, Problem 15E

Chapter
Section
Textbook Problem

Find the limit or show that it does not exist. lim x → ∞ 3 x − 2 2 x + 1

To determine

To find: The value of limx3x22x+1.

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

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

Calculation:

Obtain the value of the function as x approaches infinity.

Consider f(x)=3x22x+1.

Divide both the numerator and the denominator by the highest power of x in the denominator. That is, x0

f(x)=3x2x2x+1x=3xx2x2xx+1x=32x2+1x

Take the limit of f(x) as x approaches infinity

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