   Chapter 2.6, Problem 34E

Chapter
Section
Textbook Problem

Find the limit or show that it does not exist. lim x → − ∞ 1 + x 6 x 4 + 1

To determine

To show: The value of limx1+x6x4+1 does not exist.

Explanation

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

Proof:

Consider the function f(x)=1+x6x4+1.

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

f(x)=1+x6x4x4+1x4=1x4+x6x4x4x4+1x4=1x4+x21+1x4

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

limx1+x6x4+1=limx1x4+x21+1x4

As x goes to negative infinity, 1x4+x2 goes to infinity. That is,

limx(1x4+x2)=limx(1x4)+limx(x2)=0+

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