   Chapter 2, Problem 16RE

Chapter
Section
Textbook Problem

Find the limit. lim x → − ∞ 1 − 2 x 2 − x 4 5 + x − 3 x 4

To determine

To find: The value of limx12x2x45+x3x4.

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 such that xr is defined, then limx1xr=0.

Calculation:

Obtain the value of the function as x approaches infinity.

Consider the function f(x)=12x2x45+x3x4.

Divide both the numerator and the denominator by the highest power of x in the denominator. That is, x40.

f(x)=12x2x4x45+x3x4x4=1x42x2x4x4x45x4+xx43x4x4=1x42x215x4+1x33

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

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