   Chapter 2.6, Problem 31E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find the limit or show that it does not exist. lim x → ∞ x 4 − 3 x 2 + x x 3 + x + 2

To determine

To show: The value of limxx43x2+xx3x+2 does not exist.

Explanation

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

Proof:

Consider the function f(x)=x43x2+xx3x+2.

Divide both the numerator and the denominator by the highest power of x in the denominator. That is, x30.

f(x)=x43x2+xx3x3x+2x3=x4x33x2x3+xx3x3x3xx3+2x3=x3x+1x211x2+2x3

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

limxx43x2+xx3x+2=limx(x)(3x)+(1x2)1(1x2)+(2x3)

Here, as x goes to infinity, x3x+1x goes to infinity. That is, the second and third term becomes zero as x approaches zero but the first term approaches infinity

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